fiKiii 


IN  MEMORIAM 
FLORIAN  CAJORI 


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ELEMENTS  OF  ALGEBRA 


BY 


WOOSTEE  WOODRUFF  BEMAN 

Professok  of  Mathematics  in  the  University  of  Michigan 


DAVID   EUGENE    SMITH 

Principal  op  the  State  Normal  School  at  Brockport,  N.Y. 


BOSTON,  U.S.A. 
GINN  &  COMPANY,  PUBLISHERS 

1900 


Copyright,  1900,  by 
WoosTEB  Woodruff  Beman  and  David  Eugene  Smith 


ALL  RIGHTS   RESERVED 


CAJQIU: 


PREFACE. 


In  the  preparation  of  this  work  the  authors  have  followed  their 
usual  plan  of  attempting  to  allow  the  light  of  modern  mathe- 
matics to  shine  in  upon  the  old,  and  to  do  this  by  means  of  a 
text-book  which  shall  be  usable  in  American  high  schools,  acade- 
mies, and  normal  schools. 

In  general,  the  beaten  paths  have  been  followed,  experience 
having  developed  these  and  having  shoVgti  their  safety  and  value. 
But  where  there  is  an  unquestionable  gain  in  departing  from 
these  paths  the  step  has  been  taken.  For  example,  the  subject 
of  factoring  has  recently  attracted  the  attention  it  deserves ;  in 
fact,  several  writers  have  carried  it  to  an  unjustifiable  extreme  ; 
but  there  are  few  text-books  that  mention  the  subject  after  the 
chapter  is  closed  ;  it  is  taught  with  no  applications,  and  the  stu- 
dent is  usually  left  with  the  idea  that  it  has  none.  The  authors 
have  departed  from  this  plan,  and  have  followed  the  chapter  with 
certain  elementary  applications,  using  the  method  in  solving  easy 
quadratic  and  higher  equations,  ^making'  much  use  of  it  in  frac- 
tions, and  not  ceasing  to  review  it  and  its  applications  until  it 
has  come  to  be  a  familiar  and  indispensable  tool.  By  following 
such  a  scheme  the  student  knows  much  of  quadratics  before  he 
reaches  the  chapter  on  the  subject,  and  he  enters  upon  it  with 
increased  intelligence  and  confidence. 

The  arrangement  of  chapters  has  been  the  subject  of  consider- 
able experiment  of  late.  But  the  plan  adopted  in  this  work  is, 
in  general,  based  upon  the  following : 

1.  The  new  should  grow  out  of  the  old,  as  the  expressions  of 
algebra  out  of  those  of  arithmetic,  the  negative  number  out  of 
familiar  concepts,  factors  out  of  elementary  functions,  quadratic 
and  higher  equations  out  of  factoring,  the  theory  of  indices  out 
of  the  three  fundamental  laws  for  positive  integral  indices,  the 
complex  number  out  of  the  surd,  and  so  on. 

iii 


iv  PREFACE. 

2.  The  student's  interest  should  be  excited  as  early  as  possible, 
and  it  should  be  maintained  by  reviews  and  by  applications  to 
modern  concrete  problems.  To  this  end  the  equation  has  been 
introduced  in  the  first  chapter,  with  simple  applications,  and 
general  review  exercises  have  been  inserted  at  frequent  intervals. 

3.  The  new  should  be  introduced  where  it  is  needed.  To  put 
the  remainder  theorem  where  it  is  usually  placed,  at  the  end  of 
the  work,  is  entirely  unwarranted  ;  it  is  needed  just  before  fac- 
toring. To  put  complex  numbers  after  quadratics  is  equally  unsci- 
entific, for  they  are  met  on  the  very  threshold  of  this  subject. 

Considerable  attention  has  been  given  to  the  illustration  of 
algebraic  laws  by  graphic  forms.  The  value  of  this  plan  is  evi- 
dent ;  the  picture  method,  the  coordination  of  the  concrete  and 
the  abstract,  the  one-to-one  correspondence  between  thought  and 
thing  —  this  has  been  recognized  too  long  to  require  argument. 
This  method  of  making  algebraic  abstractions  seem  real  is  fol- 
lowed in  the  presentation  of  certain  fundamental  laws  (p.  37), 
in  the  study  of  certain  common  products  (p.  51),  but  more  espe- 
cially in  the  treatment  of  the  complex  number  (p.  236)  —  a  subject 
usually  passed  with  no  understanding,  —  and  (in  the  Appendix) 
in  the  study  of  equations. 

Where  the  time  and  the  maturity  of  the  class  allow,  the  Appen- 
dix may  profitably  be  studied  in  connection  with  the  several  chap- 
ters to  which  it  refers.  This  arrangement  allows  the  teacher  to 
cover  the  usual  course,  or  to  make  it  somewhat  more  elaborate  if 
desired. 

It  need  hardly  be  said  that  no  class  is  expected  to  solve  more 
than  half  of  the  exercises,  the  large  number  being  inserted  to 
allow  of  a  change  from  year  to  year. 

It  is  the  hope  of  the  authors  that  their  efforts  to  prepare  a  text- 
book adapted  to  American  schools  of  the  twentieth  century  may 
meet  the  approval  of  teachers  and  students.  It  is  believed  that 
they  have  lessened  the  general  average  of  difficulty  of  the  old- 
style  text-book,  while  greatly  adding  to  the  mathematical  spirit. 


June  1.  1900. 


W.  W.  BEMAN,  Ann  Arbor,  Mich. 
D.  E.  SMITH,  Brockport,  N.  Y. 


TABLE  OF  CONTEE^TS. 


CHAPTER  I. 
INTRODUCTION  TO  ALGEBRA. 

PAGE 

I.     Algebraic  Expressions 1 

II.     The  Equation 8 

III.  The  Negative  Number 17 

IV.  The  Symbols  of  Algebra 21 

V.     Propositions  of  Algebra 25 

CHAPTER   II. 
ADDITION  AND  SUBTRACTION. 

T.     Addition 27 

II.     Subtraction 31 

III.  Symbols  of  Aggregation .  35 

IV.  Fundamental  Laws 37 

CHAPTER   III. 
MULTIPLICATION. 

I.     Definitions  and  Fundamental  Laws 40 

II.     Multiplication  of  a  Polynomial  by  a  Monomial          .         .  45 

III.     Multiplication  of  a  Polynomial  by  a  Polynomial        .         .  46 


VI  TABLE   OF   CONTENTS. 


PAOE 

IV.     Special  Products  Frequently  Met 51 

V.     Involution .         .       53 


CHAPTER    IV. 
DIVISION. 
I.     Definitions  and  Laws         .......       60 

II.     Division  of  a  Polynomial  by  a  Monomial  ....       61 

III.     Division  of  a  Polynomial  by  a  Polynomial         ...       63 

CHAPTER   V. 

ELEMENTARY  ALGEBRAIC  FUNCTIONS. 

I.     Definitions 69 

II.     The  Remainder  Theorem 74 

CHAPTER   VI. 
FACTORS. 

I.     Types '78 

II.     Application  of  Factoring  to  the  Solution  of  Equations      .       91 
III.     Evolution .92 

CHAPTER   VII. 

HIGHEST    COMMON   FACTOR   AND    LOWEST    COMMON 
MULTIPLE. 

I.     Highest  Common  Factor 103 

II.     Lowest  Common  Multiple 110 

CHAPTER  VIII. 

FRACTIONS. 

Definitions .         .         .     114 

I.     Reduction  of  Fractions 115 


TABLE   OF   CONTENTS.  vil 


PAOE 

II.  Addition  and  Subtraction  .....  123 

III.  Multiplication 126 

IV.  Division 132 

V.  Complex  Fractions 135 

VI,     Fractions  of  the  form  ?; »    t:  ^    35 140 


Y 


CHAPTER   IX. 


SIMPLE   EQUATIONS    INVOLVING    ONE    UNKNOWN 
QUANTITY. 

I.     General  Laws  Governing  the  Solution       ....  145 

II.     Simple  Integral  Equations 153 

III.  Simple  Fractional  Equations 154 

IV.  Irrational  Equations  Solved  like  Simple  Equations   .         .  160 
V.     Application  of  Simple  Equations 163 


M       CHAPTER  X. 


SIMPLE    EQUATIONS   INVOLVING   TWO    OR   MORE 
UNKNOWN    QUANTITIES. 


Definitions 

I.     Elimination  by  Addition  or  Subtraction    . 
II.     Elimination  by  Substitution  and  Comparison 

III.  General  Directions 

IV.  Applications,  Tw^o  Unknown  Quantities   . 

V.     Systems   of   Equations    with   Three   or   More 
Quantities 

VI.     Applications,  Three  Unknown  Quantities 


Unknown 


179 

180 
183 
186 
189 

192 
197 


CHAPTER   XI. 
INDETERMINATE  EQUATIONS.  .  .     201 


VIU  TABLE   OF   CONTENTS. 

CHAPTER    XII. 
THE  THEORY  OP  INDICES. 

PAGK 

I.     The  Three  Fundamental  Laws  of  Exponents     .  .     204 

II.     The  Meaning  of  the  Negative  Integral  Exponent       .         .     205 

III.  The  Meaning  of  the  Fractional  Exponent  .         .         ,     208 

IV.  The  Three  Fundamental  Laws  for  Fractional  and  Negative 

Exponents 212 

V.     Problems  Involving  Fractional  and  Negative  Exponents   .     216 
VI.     Irrational  Numbers.     Surds      ......     220 

VII.     The  Binomial  Theorem 233 

CHAPTER   XIII. 

COMPLEX  NUMBERS. 

I.     Definitions 236 

II.     Operations  with  Complex  Numbers 242 

CHAPTER   XIV. 

QUADRATIC  EQUATIONS  INVOLVING  ONE  UNKNOWN 
QUANTITY. 

I.  Methods  of  Solving 246 

II.  Discussion  of  Roots 260 

III.  Equations  Reducible  to  Quadratics 266 

IV.  Problems  Involving  Quadratics  .         .         .         .         .  276 

CHAPTER   XV. 
SIMULTANEOUS  QUADRATIC  EQUATIONS. 
I.     Two  Equations  with  Two  Unknown  Quantities         .         .     284 
II.     Three  or  More  Unknown  Quantities  ....     298 

III.     Problems  Involving  Quadratics  .....     300 


TABLE   OF    CONTENTS.  IX 

CHAPTER    XVL 

PAGK 

INEQUALITIES.    MAXIMA  AND  MINIMA.  .     304 

CHAPTER    XVII. 
RATIO,  VARIATION,  PROPORTION. 

I.     Ratio 310 

II.     Variation 319 

III.     Proportion         .         ...         .         .         -^        •         •         •     326 

CHAPTER   XV III. 
SERIES. 

Definitions 334 

I.     Arithmetic  Series 335 

II.     Geometric  Series       ........  340 

III.     Miscellaneous  Types 346 

CHAPTER    XIX. 
LOGARITHMS 348 

CHAPTER    XX. 
PERMUTATIONS  AND  COMBINATIONS.  .      364 

CHAPTER   XXI. 
THE  BINOMIAL  THEOREM.      .  .373 


APPENDIX. 

I.     Proof  of  the  Binomial  Theorem  for   Positive  Integral 

Exponents  (p.  57) 377 

II.     Synthetic  Division  (p.  67) 378 


X  TABLE    OF   CONTENTS. 

PAOE 

III.  The  Applications  of  Homogeneity,  Symmetry,  and  Cyclo- 

Symmetry  (p.  73) 380 

IV.  Application  of  the  Laws  of  Symmetry  and  Homogeneity 

to  Factoring  (p.  88) 387 

V.     General  Laws   Governing   the    Solution    of    Equations 

(p.  152) 390 

VI.     Equivalent  Systems  of  Equations  (p.  185)        .         .         .  394 

VII.     Determinants  (p.  198) 395 

VIII.     Graphic  Representation  of  Linear  Equations  (p.  202)      .  408 

IX.     Graphs  of  Quadratic  Equations  (p.  296)  ....  415 


TABLES. 

Table  of  Biographies 423 

Table  of  Etymologies 427 


ELEMENTS  OF  ALGEBRA. 

CHAPTER   I. 

INTRODUCTION   TO    ALGEBRA. 

I.    ALGEBRAIC  EXPRESSIONS. 

1.  There  is  no  dividing  line  between  the  arithmetic 
with  which  the  student  is  familiar  and  the  algebra  which 
he  is  about  to  study.  Each  employs  the  symbols  of  the 
other,  each  deals  with  numbers,  each  employs  expressions 
of  equality,  and  each  uses  letters  to  represent  numbers. 

In  arithmetic  the  student  has  learned  the  meaning  of  2^ ; 
in  algebra  he  will  go  farther  and  will  learn  the  meaning  of 
2^.  In  arithmetic  he  has  learned  the  meaning  of  3  —  2 ;  in 
algebra  he  will  go  farther  and  will  learn  the  meaning  of 
2-3. 

In  arithmetic  he  has  said, 

If  2  X  some  number  equals  10, 
the  number  must  be  ^  of  10,  or  5. 

In  algebra  he  will  express  this  more  briefly,  thus : 

If  2a;  =  10, 

then  X  =  5; 

indeed  he  may  already  have  met  this  form  in  arithmetic. 

By  arithmetic  he  probably  could  not  solve  a  problem  of 
this  nature:  The  square  of  a  certain  number,  added  to  5 
times  that  number,  equals  50;  to  find  the  number.  But 
after  studying  algebra  a  short  time,  he  will  find  the  solu- 
tion quite  simple. 

1 


2  ..    ,  __      ^     ELEjMENTS   OF  ALGEBRA. 

In  arithmetic  it  is  quite  common  to  use  a  letter  to  repre- 
sent a  number,  as  r  to  represent  the  rate  of  interest,  i  to 
represent  the  interest  itself,  p  the  principal,  etc.  In 
algebra  this  is  much  more  common.  In  arithmetic  it  is 
customary  to  denote  multiplication  by  the  symbol  x ,  the 
product  of  5%  and  $100  being  written  5%  x  $100,  and 
the  product  of  r  and  phj  rxp\  but  in  algebra  the  latter 
product  is  represented  by  rp. 

In  expressing  5  times  2  we  cannot  write  it  52,  because  that  means 
50  +  2.  Bat  where  only  letters  are  used,  or  one  numeral  and  one  or 
more  letters,  we  may  define  the  absence  of  a  sign  to  mean  multiplica- 
tion. Thus,  ah  means  a  x  6,  that  is,  the  product  of  the  numbers  rep- 
resented by  a  and  h;  bah  means  5  times  this  product. 

EXERCISES.    I. 

If  ct  =  5,  b  =  7,  c  =  3,  d  =  1,  e  =  4,  find  the  value  of 

each  of  the  expressions  in  exs.  1-9. 

^    J  ^  o       7  21  ae 

1.    oabd.  2.    facde.  3. 


4.  bed 


nrr—^ —  be  —  ad  /. 


35  ab 
23c  "■    ^3cde' 


b  cd 

If  a  =  2,  b  =  S,  c  =  4:,  d  =  5,  find  the  value  of  each  of 
the  expressions  in  exs.  10-17. 


10. 

abe       abc 
bed       acd 

11. 

a  -{-  d      c  —  b 
lb            3 

12. 

a      b       d 
b       c       be 

13. 

4      6_8      10 
abed 

14. 

c       3d      2b 
a        b        3a 

15. 

a       ^   I    c        d 
4~6"^8""10 

16. 

a  -{-  c   1   d  —  b 

7                      1 

-3. 

17. 

a  +  b  ^c  +  d 

5a  —  d 


INTRODUCTION   TO   ALGEBRA.  3 

2.  A  collection  of  letters,  or  of  letters  and  other  number- 
symbols,  connected  by  any  of  the  signs  of  operation  (+,  — , 
X ,  H-,  etc.)  is  called  an  algebraic  expression. 

E.g. ,  3  X  +  2  a  is  an  algebraic  expression,  but  3  +  2  is  an  arithmet- 
ical expression.  So  2  a  is  an  algebraic  expression,  2  and  a  being 
connected  by  the  (understood)  sign  of  multiplication;  also  a,  since 
that  means  1  a. 

3.  An  algebraic  expression  containing  neither  the  +  nor 

the  —  sign  of  operation  is  called  a  term  or  a  monomial. 

T^       c    -L    r    I —    lOaftic  .  ,       ^      , 

E.g..,  ^ao.,  5  Vox,  -—z ,  are  monomials.     In  the  expression  2  aic 

£i6  yz 

+  Sby  —  ISy^,  the  expressions  2 ax,  3 by,  and  5 y^,  are  the  terms,  and 
each  taken  by  itself  is  called  a  monomial.  The  broader  use  of  the 
word  term  is  given  in  §  46. 

4.  An  algebraic  expression  made  up  of  several  terms  or 
numbers  connected  by  the  sign  +  or  —  is  called  a  polynomial. 

The  word  means  many-termed.  On  all  such  new  words  consult 
the  Table  of  Etymologies  in  the  Appendix. 

5.  A  polynomial  of  two  terms  is  called  a  binomial,  one 
of  three  terms  a  trinomial.  Special  names  are  not  given 
to  polynomials  of  more  than  three  terms. 

E.g.,  -a^  —  -  is  a  binomial.     5  Va 1-  ah^cd  is  a  trinomial. 


EXERCISES.    II. 

1.  Select  the  algebraic  expressions  in  the  following  list : 

(a)  Sa%c.  (b)  ^a'bcd. 

(c)   ^  -  c'd\  (d)  x^ -{-!/  +  z\ 

(e)  2-3V7H-1.  (f)  2x^-?>x'-^x-V\. 

2.  Out  of  the  algebraic  expressions  select  the  monomials. 

3.  Out  of   the   polynomials    select   the    binomials;    tri- 
nomials. 


4  ELEMENTS   OF   ALGEBRA.  • 

6.  In  the  operation  of  multiplication  expressed  hy  ax  b 
X  c,  or  abc,  the  a,  b,  and  c  are  called  the  factors  of  the 
expression,  and  the  expression  is  called  a  multiple  of  any 
of  its  factors. 

Factors  should  be  carefully  distinguished  from  terms.  The 
former  are  connected  by  signs  of  multiplication,  expressed 
or  understood ;  the  latter  by  signs  of  addition  or  subtraction. 

7.  Any  factor  of  an  expression  is  called  the  coefficient  of 
the  rest  of  the  product.  The  word,  however,  is  usually 
applied  only  to  some  factor  whose  numerical  value  is  ex- 
pressed or  known  and  which  appears  first  in  the  product. 

E.g.^  in  the  expression  3  ax,  8  is  the  coefficient  of  ax,  and  3  a  is  the 
coefficient  of  x. 

Since  a  =  la,  the  coefficient  1  may  be  understood  before 
any  letter. 

8.  As  in  arithmetic,  the  product  of  several  equal  factors 
is  called  a  power  of  one  of  them. 

j&. ST.,  2  X  2  X  2  is  called  the  third  power  of  2  and  is  written  2^ ; 
aaaaa  is  called  the  fifth  power  of  a  and  is  written  a^. 

9.  The  number-symbol  which  shows  how  many  equal 
factors  enter  into  a  power  is  called  an  exponent. 

E.g.,  in  2^,  3  is  the  exponent  of  2 ;  in  a^,  5  is  the  exponent  of  a. 
The  exponent  affects  only  the  letter  or  number  adjacent  to  which  it 
stands  ;  thus,  ab^  means  abbb. 

The  exponent  should  be  carefully  distinguished  from  the 
coefficient.  In  the  expression  2  ax^,  2  is  the  coefficient  of 
ax^,  and  2  a  oi  x^;   3  is  the  exponent  of  x. 

Since  x  may  be  considered  as  taken  once  as  a  factor  to 
make  itself,  x^  is  defined  as  meaning  x.  Hence,  any  letter 
may  be  considered  as  having  an  exponent  1. 

There  are  other  kinds  of  powers  and  exponents  besides  tliose  which 
have  just  been  defined,  and  these  will  be  discussed  later  in  the  work. 


INTRODUCTION  TO  ALGEBRA.  6 

10.  The  degree  of  a  monomial  is  determined  by  the  number 
of  its  literal  factors. 

E.g.^  a^  is  of  the  5th  degree,  a^ft*  of  the  7th,  3  abc  of  the  3d.,  and  5  a 
of  the  1st.  A  number,  like  5,  is  spoken  of  as  of  zero  degree  because  it 
has  no  literal  factors. 

11.  The  word  degree  is  usually  limited,  however,  by  refer- 
ence to  some  particular  letter. 

Thus,  while  ^a^'-x^  is  of  the  5th  degree,  it  is  said  to  be  of  the  3d 
degree  in  x,  or  of  the  2d  degree  in  a,  or  of  zero  degree  in  other  letters. 

12.  Terms  of  the  same  degree  in  any  letter  are  called 
like  terms  in  that  letter. 

Thus,  3  ax2  and  5  ox^  are  like  terms,  being  of  the  same  degree  in 
each  letter.     3  ax^  and  5  ftx^  are  like  terms  in  x. 

13.  The  degree  of  a  polynomial  is  the  highest  degree  of 
any  of  its  terms. 

Thus,  ax2  +  6x  +  c  is  of  the  second  degree  in  x. 

14.  As  in  arithmetic,  one  of  the  two  equal  factors  of  a 
second  power  is  called  the  square  (or  second)  root  of  that 
power,  one  of  the  three  equal  factors  of  a  third  power  the 
cube  (or  third)  root,  one  of  the  four  equal  factors  of  a  fourth 
power  is  called  the  fourth  root,  etc. 

The  word  root  has  also  a  broader  meaning,  as  in  "the 
square  root  of  2,"  an  expression  which  is  legitimate,  although 
2  is  not  a  second  power  of  any  integral  or  fractional  num- 
ber.    This  meaning  will  be  discussed  later. 

The  square  root  of  a  is  indicated  either  by  Va  or  by  a^  the  cube 
root  by  Va  or  by  a^,  the  fourth  root  by  Va  or  by  a^,  etc.  In  a^,  the 
i  is  called  a  fractional  exponent,  and  the  term  is  read  "a,  exponent 
i,"  or  "the  square  root  of  a,"  or  "a  to  the  i  power,"  a  reading 
which  will  be  justified  by  the  subsequent  explanation  of  the  word 
power. 


6  ELEMENTS  Oi?  ALGEBliA. 

From  what  has  been  stated  it  will  be  seen  that  one  of  the 
features  of  algebra  is  the  representation  of  numbers  by 
letters.     The  advantages  of  this  plan  will  soon  appear. 

Thus,  if  a  number  is  represented  by  n,  5  times  the  square 
of  that  number  will  be  represented  by  5  n^.  If  two  num- 
bers are  represented  by  a  and  b,  3  times  the  cube  of  the 
first,  divided  by  5  times  the  square  root  of  the  second,  will 

3  a^              3  a^ 
be  represented  by  — -  or  by 

15.  Those  terms  of  a  polynomial  which  contain  letters 
constitute  the  literal  part  of  the  expression. 

E.g. ,  the  literal  part  of  x^  +  2  a;  +  1  is  x2  +  2  x. 
The  expression  is^  also  used  with  respect  to  factors.      Thus,   the 
literal  part  of  ia^2  is  a. 


EXERCISES.    III. 

1.  What  is  the  numerical  value  of  each  term  in  the  fol- 
lowing expressions,  it  a  =  1,  b  =  2,  c  =  5,  d  =  3  ? 

(a)  ab^c'd\  (b)  c^  +  b^-Sa. 

(0)2^-100-2,.  (d)«  +  A  +  ^  +  ^. 

2.  In  ex.  1,  what  is  the  numerical  value  of  each  poly- 
nomial ? 

3.  In  13  a%^x,  what  is  the  coefficient  of  £c  ?  of  b^x  ?  of 
a%^x  ?  What  is  the  degree  of  the  expression  ?  What  is  its 
degree  in  ic  ?     What  is  the  exponent  oi  a?    ot  b?   oi  x? 

4.  In  the  following  monomials  name  the  coefficients  of 
the  various  powers  of  x,  and  also  the  exponents  of  x : 

(a)  ^-  (b)  x^  (c)  ^x». 

(d)  23  a V.  (e)  ia^'cxK  (f)  ^a^Vbx. 


INTKODUCTION  TO   ALGEBRA.  7 

5.  From  ax^,  3  hx^,  cx^,  a^x,  and  10  ahx^,  select  the  like 
terms  in  x  or  any  of  its  powers. 

6.  From  3  ax"^,  9  mx,  14  ax^,  ax^,  9  ax^,  and  144  x,  select 
the  like  terms. 

7.  Express  algebraically  that  if  x^  +  y^  +  2xy  be  divided 
hj  x  -{-  y  the  quotient  \&  x  -\-  y.     (Use  fractional  form.) 

8.  What  is  the  degree  of  the  polynomial  ax^  -[-hx  -\-  c? 
What  is  its  degree  in  ic  ?  What  is  its  value  if  a  =  b  =  c  —  l, 
and  X  =  b? 

9.  Express  algebraically  that  if  the  sum  of  a^,  ab,  and 
b'^  be  divided  by  the  square  of  the  binomial  c  —  d,  the  quo- 
tient is  X. 

10.  What  is  the  meaning  of  the  expression 

(That  from  4  times  the  square  of  a  certain  number  there 
has  been  subtracted,  etc.) 

11.  Also  of  the  following  expressions : 
(a)  a'  +  2ab  +  b\  (b)  a^  -  b\ 

(c)  ^a^-4.b^  +  a^.  (d)  a^  +  3  a^^*  +  3  ab^  +  b\ 

12.  Eepresent  algebraically  the  sum  of  3  times  the  square 
of  a  number,  f  the  cube  root  of  a  second  number,  and  5 
times  the  5th  power  of  a  third  number.  What  is  the  value 
of  the  expression,  if  the  three  numbers  are  respectively  2, 
8,  1  ? 

13.  Given  a  =  4,  b  =  Q>,  c  =  9,  d  =  1Q>,  e  =  8,  find  the 
value  of  each  of  the  following,  and  designate  the  expres- 
sion as  a  monomial,  binomial,  etc. : 

(a)  2  a'bc^.  (b)  d^e^  -  b. 

/  \     -i-  ,   z-   ,    J   ,     2                     /JN   25  abcde 
(c)  ct*  -f  &  +  ^  +  e\  (d)  —^ 

(e)  25#-j-a2-^  +  5.  (f)  ^b^-c^J^eK 

o 


8  ELEMENTS  OF  ALGEBRA. 

II.  THE  EQUATION. 

16.  An  equality  which  exists  only  for  particular  values  of 
certain  letters  representing  the  unknown  quantities  is  called 
an  equation.  These  particular  values  are  called  the  roots  of 
the  equation. 

Thus,  X  +  3  =  5  is  an  equation  because  the  equaUty  is  true  only  for 
a  particular  value  of  the  unknown  quantity  x,  that  is,  for  x  =  2.  This 
equation  contains  only  one  unknown  quantity. 

2  +  3  =  5  expresses  an  equality,  but  it  is  not  an  equation  as  the 
word  is  used  in  algebra. 

17.  The  discovery  of  the  roots  is  called  the  solution  of  the 
equation,  and  these  roots  are  said  to  satisfy  the  equation. 

Thus,  if  X  +  5  =  9,  the  equation  is  solved  when  it  is  seen  that  x  =  4. 
This  value  of  x  satisfies  the  equation,  for  4  +  5  =  9. 

18.  If  two  algebraic  expressions  have  the  same  value 
whatever  numbers  are  substituted  for  the  letters,  they  are 
said  to  be  identicaL 

Thus,  a^  _| is  identical  to  a^  _f-  5^  and  a  +  6  to  6  +  a. 

An  identity  is  indicated  by  the  symbol  = ,  as  in  a^  +  &  =  6  +  a^. 

19.  The  part. of  an  equation  to  the  left  of  the  sign  of 
equality  is  called  the  first  member,  that  to  the  right  the 
second  member,  and  similarly  for  an  identity. 

The  two  members  are  often  spoken  of  as  "  the  left  side  "  and  "  the 
right  side,"  respectively. 

The  extensive  use  of  the  equation  is  one  of  the  character- 
istic features  of  algebra. 

The  importance  and  the  treatment  of  the  equation  will 
best  be  understood  by  considering  a  few  problems. 

In  each  case  we  say,  "  Let  x  =  the  number,"  meaning  that  x  is  to 
represent  the  unknown  quantity. 


INTRODUCTION   TO   ALGEBRA.  9 

1.  Find  the  nuviber  to  twice  which  if  3  is  added  the 
result  is  11. 

1.  Let  X  =  the  number. 

2.  Then  '2x  =  twice  the  number. 

3.  Hence,  2x  +  3  =  ll.  (Why?) 

4.  Subtracting  3  from  these  equals,  the  results  must  be  equal,  and 

2x=ll  -3,  or8. 

5.  Dividing  these  equals  by  2,  the  results  must  be  equal,  and 

x  =  4. 
Check.     To  see  if  this  value  of  x  satisfies  the  equation,  substitute  it 
in  step  3.     Since  2x4  +  3  =  11,  the  result  is  correct.     This  is  called 
checking  or  verifying  the  result. 

20.  A  check  on  an  operation  is  another  operation  whose 
result  tends  to  verify  the  result  of  the  first. 

^.gr.,  if  11  —  7  =  4,  then  4  +  7  should  equal  11 ;  this  second  result, 
11,  verifies  the  first  result,  4. 

The  secret  of  accurate  work  in  algebra  and  in  arithmetic 
lies  largely  in  the  continued  use  of  proper  checks. 

21.  A  check  on  a  solution  of  an  equation  is  such  a  substitu- 
tion of  the  root  as  shows  that  it  satisfies  the  given  equation. 

This  substitution  must  always  be  inade  in  the  original 
equation  or  in  the  statement  of  the  problem.  Thus,  in  the 
above  solution  it  would  not  answer  to  substitute  the  root, 
4,  in  step  4,  because  a  mistake  might  have  been  made  in 
getting  step  4  from  step  3. 

2.  Two-thirds  of  a  certain  number,  added  to  5,  equals  17. 
What  is  the  number  ? 

1.  Let  X  =  the  number. 

2.  Then  |-x  +  5  =  17,  by  the  conditions  of  the  problem. 

3.  Subtracting  5  from  these  equals,  the  results  must  be  equal,  and 

lx  =  \2. 

4.  Therefore,  x  =  18. 
Check.     I  of  18  =  12,  and  12  +  5  =  17. 


10  ELEMENTS   OF  ALGEBRA. 

3.  72  divided  by  a  certain  number  equals  twice  that  num- 
ber.     What  is  the  number  ? 

1.  Let  X  =  the  number. 

72 

2.  Then  —  =  twice  the  number,  by  the  conditions 

^  of  the  problem. 

3.  Therefore,  —  =  2x. 

X 

4.  Multiplying  these  equals  by  x,  the  results  must  be  equal,  and 

72  =  2x2. 

5.  Dividing  these  equals  by  2, 

36  =  x2. 

6.  Extracting  the  square  roots  of  these  equals, 

Q  =  x. 


4.  If  from  35  a  certain  number  is  subtracted,  the  differ- 
ence equals  the  sum  of  twice  that  number  and  20,  What  is 
the  number  ? 


.1. 

Let 

X  =  the  number. 

2. 

Then 

35  -  X  =:=  2  X  +  20. 

(Why  ?) 

3. 

Then 

35  =  3x  +  20,  by  adding  x. 

4. 

Then 

15  =  3x. 

(Why  ?) 

5. 

Then 

5  =  X. 

(Why?) 

Check.     (What  should  it  be  ?) 

From  the  preceding  problems  it  will  be  seen  that  the  two 
members  of  an  equation  are  like  the  weights  in  two  pans  of 
a  pair  of  scales  which  balance  evenly ;  if  a  weight  is  taken 
from  one  pan,  an  equal  weight  must  be  taken  from  the  other 
if  the  even  balance  is  preserved ;  if  a  weight  is  added  to  one 
pan,  an  equal  weight  must  be  added  to  the  other ;  and,  in 
general,  any  change  made  in  one  side  requires  a  like  change 
in  the  other. 

These  facts  are  already  known  from  arithmetic,  where  the  equation 
is  frequently  met.  Even  in  primary  grades  problems  are  given  like 
2  X  (?)  =  12,  this  being  merely  an  equation  with  the  symbol  (?)  in  place 
of  X. 


INTRODUCTION   TO  ALGEBRA.  11 

22.  The  axioms.  There  are  several  general  statements 
(of  which  a  few  have  already  been  used)  so  obvious  that 
their  truth  may  be  taken  for  granted.  Such  statements 
are  called  axioms. 

The  following  are  the  axioms  most  frequently  met  in 
elementary  algebra. 

1.  Quantities  which  are  equal  to  the  same  quantity,  or  to 
equal  quantities,  are  equal  to  each  other. 

That  is,  if  5  —  X  =  3,  and  1  +  x  =  3,  then  5  —  x  =  1  +  x. 

2.  If  equals  are  added  to  equals,  the  sums  are  equal. 
That  is,  \tx  =  y,  then  x-\-2  =  y  -\-2. 

3.  If  equals  are  subtracted  from,  equals,  the  remainders 
are  equal. 

That  Is,  if  X  +  2  =:  9,  then  x  =  9  -  2,  or  7. 

4.  If  equals  are  added  to  unequals,  the  sums  are  unequal 
in  the  same  sense. 

"  In  the  same  sense  "  means  that  if  the  first  was  greater  than  the 
second  before  the  addition  of  the  equals,  it  is  after.  Thus,  if  x  is 
greater  than  8,  x  +  2  is  also  greater  than  10. 

5.  If  equals  are  subtracted  from  unequals,  the  remainders 
are  U7iequal  iii  the  same  sense. 

That  is,  if  x  is  less  than  16,  x  —  3  is  less  than  13. 

6.  If  equals  are  multiplied  by  equal  numbers,  the  prod- 
ucts are  equal. 

That  is,  if  -  =  6,  x  =  3  x  6,  or  18. 
3 

7.  If  equals  are  divided  by  equals,  the  quotients  are  equal. 
That  is,  if  2  X  =  6,  x  =  6  -r-  2,  or  3. 

8.  Like  powers  of  equal  numbers  are  equal. 

That  is,  if  X  =  5,  x2  =  25.  We  here  speak  of  x  as  a  number  because 
it  represents  one. 


12  ELEMENTS   OF  ALGEBRA. 

9.  Like  roots  of  equal  numbers  are  arithmetically  equal. 

That  is,  if  a;2  =  36,  X  =  6.  The  axiom  says  "  arithmetically  equal," 
because  it  will  soou  be  found  that  there  is  an  algebraic  sense  in  which 
roots  require  special  consideration. 

These  axioms  should  at  once  be  learned  by  number. 

23.  Stating  the  equation.  The  greatest  difficulty  experi- 
enced by  the  student  in  the  solution  of  problems  is  in  the 
statement  of  the  conditions  in  algebraic  language.  After 
the  equation  is  formed  the  solution  is  usually  simple. 

While  there  is  no  method  applicable  to  all  cases,  the  fol- 
lowing questions  usually  lead  the  student  to  the  statement : 

1.  What  shall  x  represent?  In  general,  x  represents  the 
number  in  question. 

E.g.,  in  the  problem,  "Two-thirds  of  a  certain  number,  plus  10, 
equals  30,  what  is  the  number  ?  "     x  represents  the  number. 

2.  For  what  number  described  in  the  problem  may  two 
expressions  be  found? 

Thus,  in  the  above  problem,  30  and  "f  of  a  certain  number,  plus 
10,"  are  two  expressions  for  the  same  number. 

3.  How  do  you  state  the  equality  of  these  expressions  in 
algebraic  language  ? 

fx-H0  =  30. 

EXERCISES.    IV. 

Form  the  equations  for  the  following  problems : 

1.  The  difference  of  two  numbers  is  14  and  the  smaller 
is  3.     What  is  the  larger  ? 

2.  A's  money  is  three  times  B's,  and  together  they  have 
$364.      How  much  has  B? 

3.  The  sum  of  two  numbers  is  60  and  the  difference  is 
40.     What  is  the  smaller  number  ? 


INTRODUCTION  TO   ALGEBRA.  13 

Typical  solutions.  In  the  solution  of  problems  involving 
equations,  the  axioms  need  not  be  stated  in  full  except  when 
this  is  required  by  the  teacher.  The  check  (which  is  a 
complete  verification)  should  always  be  given  in  full,  except 
when  the  teacher  directs  to  the  contrary.  The  following 
solutions  may  be  taken  as  types : 

1.    What  is  that  number  to  whose  square  root  if  2  is  added 


the  result  is 

7? 

L    Let 

2.  Then 

3.  .-. 

4.  .-. 

X  =  the  number. 
Vx  +  2  =  7,  by  the  conditions. 

x  =  26. 

Ax.  3 
Ax.  8 

Check.     V25  +  2  =  5  +  2  =  7. 

2.    What  is  that  number  from  two-thirds  of  which  if  5  is 
subtracted  the  result  is  10  ? 

1.  Let  X  =  the  number. 

2.  Then  f  x  —  5  =  10,  by  the  conditions. 

3.  .-.  |-x-5  +  5  =  15,  orfx=  15.  Ax.  (?) 

4.  .-.  x  =  22^.  Ax.  (?) 
Check,     f  of  22i  =  15,  and  15  -  5  =  10. 

3.  Find  the  value  of  x  in  the  equation  Vx  +  1=^-1-7. 

1.  \^  +  1  =  ^  +  7.  Given 

2.  .-.  Vi  =  6h  or  V-.  Ax.  3 

3.  .-.  x  =  ^fi,  or  40f  Ax.  8 
Check.     (Give  it.) 

4.  Find  the  value  of  x  in  the  equation  5x  —  3  =  x  +  7. 

1.  5x  — 3  =  x  +  7.  Given 

2.  .-.  5x  =  X  +  10.      (Why  ?  See  ex.  2,  step 3) 

3.  .-.  4  X  =  10,  for  5  X  —  X  means  5  x  —  1  x. 

4.  .-.  x  =  2i.                                         (Why?) 
Check.  (Give  it.) 


14  •  ELEMENTS  OF  ALGEBKA. 


EXERCISES.    V. 

1.  Find  the  value  of  x  in  the  equation  2  £c  -f  2  =  30  +  cc. 

«Ai-20^  ^      ..      .     2      X 

2.  Also  m  —  =  5.  3.    Alsoin-  =  -- 

X  X      Z 

4.  Also  in  a;2  +  7  =  88.  5.    Also  in  a;^  -  1  =  35. 

6.  Also  in  f  X  +  5  ^  -^  ic  +  20. 

7.  Alsoin.22cc  +  30  =  17:z;  +  70. 

8.  Also  in  250  ic  -  20  =  20  :c  +  440. 

9.  Also  in  12.75  a;  +  6.25  =  7.25  x  +  17.25. 

10.  What  number  is  that  which  divided  by  3  equals  ^  ? 

11.  What  is  the  number  whose  half  added  to  16  equals 
21? 

12.  What  is  the  number  whose  tv/entieth  part  added  to 
10  equals  20  ? 

13.  What  is  that  number  to  whose  square  if  5  is  added 
the  result  is  41  ? 

14.  What  is  that  number  to  whbse  square  root  if  5  is 
added  the  result  is  41  ? 

15.  What  is  that  number  from  one-third  of  which  if  27 
is  subtracted  the  result  is  5  ? 

16.  There  is  a  number  by  which  if  9  is  divided  the  quo- 
tient is  that  number.     Find  it. 

17.  The  sum  of  a  certain  number  and  9  is  equal  to 
the  sum  of  1  and  three  times  that  number.  Find  the 
number. 

18.  The  sum  of  a  certain  number,  twice  that  number, 
and  twice  this  second  niunber-,  is  70.  What  is  the  first 
number  ? 

19.  The  united  ages  of  a  father  and  son  amount  to  100 
years,  the  father  being  40  years  older  than  the  son.  What 
is  the  asre  of  the  son  ? 


INTRODUCTION  TO  ALGEBRA.  16 

Practical  applications.  The  equation  offers  a  valuable 
method  for  solving  many  practical  problems,  of  which  a 
few  types  will  now  be  considered. 

1.  What  sum  of  money  placed  at  interest  for  1  year  at 
4^%  amounts  to  $836  ? 

1.  Let  X  =  the  number  of  dollars. 

2.  Then  x  +  0.04ix  =:  the  number  of  dollars  in  the  prin- 
•  cipal  +  the  interest. 

3.  But  836  =  the  number  of  dollars  in  the  prin- 

cipal +  the  interest. 

4.  .-.  x  +  0.0^x  =  8S6. 
6.    Or  1.04^05  =  836. 

6.  .-.  x  =  800.  Ax.  7 

7.  .-.  the  sum  is  |800.  ■ 

Check.        800  +  0.04^  of  800  =  836. 

It  should  be  noticed  that  since  x  stands  for  the  number 
of  dollars,  when  it  is  found  that  x  =  800  it  is  known  that 
the  result  is  $800. 

In  the  applied  problems  of  algebra,  x  is  always  taken  to 
represent  an  abstract  number,  and  the  first  step  should 
always  state  definitely  to  what  this  abstract  number  is 
to  refer. 

2.  A  commission  merchant  sold  some  produce  on  a  com- 
mission of2^Q,  and  paid  %%>  for  freight  and  cartage,  remit- 
ting $117.50.     For  how  much  did  he  sell  the  produce  ? 

1.  Let  X  =  the  number  of  dollars  received. 

2.  Then  x  —  0.02  x  =  the  number  after  deducting  2%. 

3.  And  x  —  0.02x  —  5  =  the  number    after   deducting   for 

cartage  also. 

4.  .-.  x-0.02x-5  =  117.50. 

5.  .-.  0.98x  =  122.60.  (Why?) 

6.  .-.  X  =  125.  (Why  ?) 
Check.     125-0.02of  125-5  =  117.50. 


16  ELEMENTS   OF   ALGEBRA. 

3.  After  deducting  -jJ^  and  then  ^  from  a  certain  sum 
there  remains  $49.50.     Required  the  sum. 

1.  Let  X  =  the  number  of  dollars. 

2.  Then  x  —  j\j  x  =  j%  x,  the  number  of  dollars  after 

deducting  jL. 

3.  From  this  ^^  x  is  to  be  taken  \  of  it, 

x^^x-iof  x%x=ifx-/^x 

4.  .-,  fx  =  49.50. 

5.  .-.  x  =  49.50 -I 

=  m. 

.-.  the  sum  is  $66. 
Check.     66  -  j\  of  66  =  59.40.         59.40  -  i  of  59.40  =  49.50. 

EXERCISES.    VI. 

1.  In  how  many  years  will  $100  double  itself  at  5% 
interest  ? 

2.  What  sum  of  money  put  at  interest  for  2  years  at  6% 
amounts  to  $84  ? 

3.  In  how  many  years  will  a  sum  of  money  double  itself 
at  6%  simple  interest  ? 

4.  In  how  many  yeai's  will  $80  amount  to  $200,  at  6% 
interest  ?     (80  +  (k  x  6%  of  80  =  200.) 

5.  What  is  the  rate  per  cent  of  premium  for  insuring  a 
house  for  $2000,  when  the  premium  is  $30  ? 

6.  Taking,  the  number  of  units  of  area  of  a  circle  as 
being  3^  times  the  square  of  the  number  of  units  of  length 
in  the  radius,  find  the  radius  of  the  circle  whose  area  con- 
tains 77|  units. 

7.  After  selling  some  goods  on  5%  commission,  a  mer- 
chant remits,  as  the  net  proceeds,  $79.80.  How  much  is 
his  commission  ?  (Let  x  =  the  number  of  dollars  for  which 
the  goods  were  sold ;   after  finding  x  take  5%  of  it.) 


INTRODUCTION  TO  ALGEBRA.  17 

III.  THE  NEGATIVE  NUMBER. 

24.  In  remote  times  men  could  count  only  by  what 
are  often  called  natural  numbers,  that  is,  1,  2,  3,  4,  5,  •••. 
Such  numbers  suffice  to  solve  an  equation  like  a;  —  3  =  0, 
an  equation  in  which  x  must  evidently  be  3. 

Mankind  then  introduced  the  unit  fraction,  that  is,  a 
fraction  with  the  numerator  1.  Such  numbers  are  neces- 
sary in  solving  an  equation  like  2  cc  —  1  =  0.     (Solve  it.) 

Then  came  the  common  fraction  with  any  numerator,  as 
§j  fj  TT'  ••••  Such  numbers  are  necessary  in  solving  an 
equation  like  3  cc  —  2  =  0.     (Solve  it.) 

The  idea  of  number  was  then  enlarged  to  cover  the  cases 
of  V2,  Vt,  v5,  •  •  •,  which  are  neither  integers  nor  fractions 
with  integral  terms.  Such  numbers  are  necessary  in  solv- 
ing an  equation  like  £c^  —  2  =  0.     (Solve  it.) 

25.  Many  centuries  later  the  necessity  was  felt  for  fur- 
ther enlarging  the  idea  of  number  in  order  to  solve  an 
equation  like  x  +  l  =  0,  oric  +  6t  =  0,  a  being  one  of 
the  kinds  of  number  above  mentioned.  This  led  to 
the  consideration  of  negative  numbers,  —  1,  —  2,  —  3,  •  •  •, 
and  the  meaning  of  these  numbers  will  now  be  inves- 
tigated. 


26.  If  the  mercury  in  a  thermometer  stands  at  5° 
above  a  fixed  point  and  then  falls  1°,  we  say  that  it 
stands  at  4°  above  that  point.  If  it  falls  another 
degree,  we  say  that  it  stands  at  3°  above  that  point, 
and  the  next  time  at  2°,  and  the  next  time  at  1°. 

If  the  mercury  then  falls  another  degree,  it  becomes 
necessary  to  name  the  point  at  which  it  stands,  and  we 
call  this  point  zero  and  designate  it  by  the  symbol  0. 

If  the  mercury  falls  another  degree,  we  must  again  name 
the  point  at  which  it  stands,  and  instead  of  calling  this 


18 


ELEMENTS  OF  ALGEBRA. 


point  "1°  below  zero,"  we  call  it  "minus  1°"  or  "negative 
1°,"  and  we  designate  it  by  the  symbol  —  1°.  Likewise,  if 
the  mercury  falls  1°  lower,  we  say  that  it  stands  at  —  2°, 
and  so  on. 


27.  Thus  we  find  a  new  use  for  the  word  minus  and  the 
symbol  — .  Heretofore  both  the  word  and  the  sign  have 
indicated  an  operation,  subtraction ;  they  now  indicate  the 
quality  of  a  number,  showing  on  which  side  of  zero  it 
stands,  and  thus  they  are  adjectives. 

In  speaking  of  "west  longitude,"  "west"  is  an  adjective  modify- 
ing "longitude";  in  speaking  of  "minus  latitude,"  "minus"  is  an 
adjective  modifying  "latitude";  so  in  "minus  2°,"  "minus"  is  an 
adjective. 

28.  It  thus  appears  that  our  idea  of  number  can 
be  enlarged  to  include  zero,  and  still  further  to 
include  the  series  of  natural  numbers  extended 
downward  from  zero. 

If  necessary  to  distinguish  1°  above  0  from  1° 
below  0,  the  former  is  written  +1°  and  called 
either  "plus  1°"  or  "positive  1°,"  and  the  latter 
is  written  —  1°.  But  unless  the  contrary  is  stated, 
a  number  with  no  sign  before  it  is  considered 
positive. 

29.  It  thus  appears  that  positive  numbers  may 
be  represented  as  standing  on  one  side  of  zero,  and 
numbers  on  the  other. 

Thus,  if  west  longitude  is  called  positive,  east  longitude 
is  called  negative,  and  vice  versa  ;  if  north  latitude  is  called 
positive,  south  latitude  is  called  negative ;  if  a  man's  capi- 
tal is  called  positive,  his  debts  are  called  negative,  etc. 

E.g.,  if  the  longitude  of  New  York  is  73°  58'  25.5''  west  and  that  of 
Berlin  is  13°  23'  43.5"  east,  the  former  may  be  designated  as  +  73°  58' 
25.5"  and  the  latter  as  -  13°  23'  43.5",  their  difference  being  87°  22'  9". 


o-f  3  or 

O-f-2  " 
0+1  .' 
0 


INTRODUCTION  TO   ALGEBRA.  19 

Similarly,  if  a  man  begins  the  year  with  $5000,  and  during  the  year 
loses  his  capital  and  gets  1 2000  in  debt,  he  is  $  7000  worse  off  than  at 
the  beginning.  It  may  then  be  said  that  he  started  with  $  5000  and 
ends  with  —  $2000,  the  difference  being  the  $7000  which  he  lost. 

30.  Since  two  such  expressions  as  +  a  and  —  a,  or  +  5° 
and  —  5",  represent  different  directions,  but  equal  measiu^es, 
they  are  said  to  have  the  same  absolute  value. 

The  symbol  |  —  a  |  is  read,  "  the  absolute  value  of  —  a." 
Hence,  I  -  5°  I  =  1  +  5°  1 ,  although  -  5°  does  not  equal  +  5°. 

Since  the  difference  between  —  5°  and  +  5°  on  a  ther- 
mometer is  10°,  it  appears  that  we  sometimes  find  the  dif- 
ference between  two  numbers  by  adding  absolute  values. 

31.  There  are  numerous  signs  used  in  algebra,  as  +,  — , 
X,  -^,  V~,  exponents,  etc.  But  by  the  sign  of  a  term  is 
always  meant  the  -{-  or  —  sign,  which  indicates  the  quality 
of  the  term,,  whether  positive  or  negative. 

Thus,  in  a^  -4-  7  6,  the  sign  of  7  6  is  plus  (understood),  while  in  aV—  7  6 
it  is  minus. 

32.  Positive  and  negative  numbers,  together  with  zero, 
are  often  called  algebraic  numbers,  positive  numbers  being 
called  arithmetical. 

Zero  is  considered  either  as  having  no  sign  or  as  having  both  the 
plus  and  the  minus  signs. 

EXERCISES.    VII. 

These  are  intended  for  oral  drill  and  should  be  supple- 
mented by  many  others  of  this  type. 

1.    A  ship  in  8°  west  longitude  (+  8°)  sails  so  as  to  lose 
1°  in  longitude.     On  what  meridian  is  it  then  ?     Suppose 
it  loses  7°  more  ?     3°  after  that  ? 
^  2.    What  is  the  difference  in  latitude  between  +  10°  and 
-  20°  ?   between  -|-  90°  and  -  90°  ? 


20  ELEMENTS   OF  ALGEBRA. 

3.  Show  that  I  5  -  7  I  =  I  -  10  +  12  I  =  I  -  22  +  20  I  =  2. 

4.  What  is  meant  by  I  —  4 1  ?    by  the  absolute  value  of 

-  8  ?   of  -  3  ? 

5.  What  is  the  absolute  value  of  10  -  17  ?  of  17  -  10  ? 
of-f?    of+l? 

6.  What  other  numbers  have  the  same  absolute  value  as 
4-  3,  -  5,  +  10,  -  V2,  0  ? 

7.  What  is  the  difference  in  time  between  50  years  b.c. 
and  50  years  a.d.  ?     Indicate  this  by  symbols. 

8.  Draw  a  line  representing  a  thermometer  scale ;  mark 
off  0°,  30°,  -  25°.     What  is  the  difference  between  30°  and 

-  25°  ? 

9.  If  the  weight  of  a  piece  of  iron  is  represented  by 
+  10  lbs.,  what  will  represent  the  weight  of  a  toy  balloon 
which  pulls  up  with  a  force  of  3  lbs.  ? 

10.  Suppose  the  piece  of  iron  and  the  balloon  mentioned 
in  ex.  9  were  fastened  together.  What  would  be  their  com- 
bined weight  ? 

11.  If  the  upward  pull  of  a  toy  balloon  is  represented  by 
+  3  lbs.,  what  will  represent  the  upward  pull  of  a  piece  of 
iron  weighing  10  lbs.  ? 

12.  What  is  meant  by  saying  that  a  person  is  worth 

-  $1000  ?     Suppose  $2000  is  added  to  his  capital.     How 
much  is  he  then  worth  ? 

13.  Draw  a  circumference  and  show  that  the  difference 
between  50°  and  -10°  equals  150°  |  +  |  -  10°  |,  or  60°. 
Also  that  the  difference  between  10°  and  -  10°  is  20°. 

14.  If  the  weights  of  two  pieces  of  iron  are  respectively 
100  lbs.  and  300  lbs.,  and  to  these  are  attached  a  balloon 
with  an  upward  pull  of  500  lbs.,  how  shall  the  combined 
weight  be  represented  ? 


INTRODUCTION  TO  ALGEBRA.  21 


IV.     THE   SYMBOLS   OF  ALGEBRA. 

33.  As  already  seen,  algebra  employs  the  symbols  of 
arithmetic,  often  with  a  broader  meaning,  and  introduces 
new  ones  as  occasion  demands.  The  following  classifica- 
tion will  enable  the  student  to  review  the  symbols  thus 
far  familiar  to  him,  and  may  add  a  few  new  ones  to  his 
list.  Others  will  be  considered  from  time  to  time  as 
needed. 

1.  Symbols  of  quantity. 

a.  Arithmetical  numbei's,  i.e.,  positive  integers  and  frac- 
tions. 

b.  Algebraic  numbers,  the  above  with  the  addition  of  neg- 
ative numbers  and  zero.     Others  will  be  considered  later. 

c.  Letters  denoting  algebraic  numbers  ;  these  are  the 
symbols  of  quantity  chiefly  used  in  algebra. 

2.  Symbols  of  quality. 

a.  The  symbols  +  a7id  —  to  indicate  positive  and  nega- 
tive number,  as  in  +  a,  —  b,  etc. 

b.  The  absolute  value  symbol,  as  in  1  —  3  | ,  indicating 
that  the  arithmetical  quality  of  —  3  is  considered. 

3.  Symbols  of  operation. 

a.  Addition,  +• 

b.  Subtraction,  — . 

c.  Multiplicatio7i,  X,  ■ ,  and  the  absence  of  sign.  Thus, 
a  xb,  a-b,  and  ab,  all  indicate  the  product  of  a  and  b.  It 
is  quite  customary  in  algebra  to  say  "  a  into  b "  for 
"  a  times  b." 

d.  Division,  -h,  /,  :,  and  the  fractional  form.      Thus, 

a  -7-  b,  a/b,  a  :  b,  and  -?  all  mean  the  quotient  of  a  divided 


hyb. 


22  ELEMENTS  OF  ALGEBRA. 

In  arithmetic  the  symbol  :  is  used  only  between  num- 
bers of  the  same  denominations ;  but  in  algebra,  where  the 
letters  represent  abstract  nmnbers,  this  distinction  does 
not  enter.  For  ease  in  typesetting  the  symbol  /  is  often 
used  in.  print ;  in  writing,  the  fraction  is  usually  employed. 

e.  Involution  and  evolution  are  indicated  by  exponents. 
Evolution  is  also  indicated,  as  in  arithmetic,  by  the  symbol 
V?  a  contraction  of  r,  the  initial  of  radix  (Latin,  root). 
Thus,  ^3  i^^eans  aaa, 

8^  means  one  of  the  three  equal  factors  of  8,  or  2. 

4.  Symbols  of  relation. 

a.  Equality,  =. 

b.  Identity,  =  ;  thus,  a  =  a,  read  "  a  is  identical  to  a." 
Also  read  "  stands  for,"  as  in  r  =  rate,  F  =  x^  -\-  2  xy,  etc. 

.c.  Inequality :  >  greater  than,  <  less  than,  ^  not  equal 
to,  >  not  greater  than,  <  not  less  than. 

5.  Symbols  of  aggregation. 

The  expression  m  (a  -\-  b)  means  that  «  +  ^  is  to  be  mul- 
tiplied by  7n.  The  parenthesis  about  a  -{-  b  is  called  a 
symbol  of  aggregation. 

The  bar,  brackets,  and  braces  are  also  used,  as  in 


a 


m  \a  —  [b  -\-  X  (a  —  b  —  c)  -{-  xa']  —  d\,  and  in 

x'^  +  2a  x  +  c^  =  {a  +  b)x'^  +  {2a-b  +  G)x  +  G'^ 

-b 

+  c 

but  the  term  parenthesis  is  often  employed  to  mean  any 
symbol  of  aggregation.  The  subject  is  more  fully  dis- 
cussed on  p.  35. 

6.    Symbols  of  deduction. 

•.',  since. 
.".,  therefore. 


INTRODUCTION   TO   ALGEBRA.  23 

7.    Symbol  of  continuation. 

•  •  • ,  meaning  "  and  so  on,"  as  in  the  sentence,  "  consider 
the  quantities  a,  a^,  a^,  •  •  • ." 

34.  Conventional  order.  Mathematicians  have  established 
a  custom  as  to  the  order  in  which  these  signs  shall  be  con- 
sidered when  several  are  involved,  as  in  an  expression  like 

a  ■\-  h  ^  c  ^  d  ^  ef^  -  g  ^  hk^  -^' 

In  the  above  expression  six  operations  are  involved,  as 
follows  : 

Direct.  Interse. 


Class     I.     Addition. 
Class    II.     Multiplication. 
Class  III.     Involution  (Powers). 


Subtraction. 
Division. 
Evolution  (Roots). 


The  mathematical  custom  is  expressed  in  the  following 
conventions  : 

1.  If  tivo  or  more  operations  of  the  same  class  come 
together  (^without  symbols  of  aggregation),  the  operations 
are  to  he  performed  in  the  order  indicated. 

E.g.,  2  +  3-4  +  1=2,  and  2x8-=-4x2  =  8. 

2.  If  two  or  more  operations  of  different  classes  come 
together  (without  symbols  of  aggregation),  the  operations 
of  the  higher  class  are  to  be  performed  first. 

I.e.,  involution  and  evolution  precede  multiplication  and  division, 
and  these  precede  addition  and  subtraction. 

^.t/.,  5  +  2  X  8 -- 22  -  Vs  =  7. 

This  conventional  order  can,  of  course,  be  varied  by  the 
use  of  symbols  of  aggregation. 

E.g.,  2  +  3  X  5  =  17,  but  (2  +  3)  X  5  =  25. 


24  ELEMENTS   OF   ALGEBRA. 

There  are  also  certain  exceptions  to  this  conventional 
order,  but  they  are  not  of  a  natui'e  to  cause  any  confusion. 

E.g.,,  ah  -r-  cd  means  {ah)  -i-  (cd)  and  not ,  and  similarly  in  other 

cases  of  the  absence  of  sign  where  division  is  involved. 

Similarly,  when  the  sign  of  ratio  (:)  appears  in  a  proportion  it  has 
not  the  same  weight  as  the  symbol  -^ .  Thus,  2  +  3:12  —  2  =  1:2 
means  (2  +  3)  :  (12  -  2)  =  1  : 2. 


EXERCISES.    VIII. 

1.  li  a  =  1,  h  =  2j  c  =  ^,  d  =  4:,  find  the  value  of  each 
of  the  following  expressions  : 

(a)  {a  +  b'^f.  (b)  b(c-\-dy. 

(c)  5d/bc-  a.  (d)  (#  -  c?  -=-  b)  c\ 

{q)  3a  +  b  X  c-d.  (f )  (^  +  *)  (p  +  d). 

(g)  2  +  a^H^  H-a  +  5.  (h)2axb-^dxc-a. 

2.  Read  the  following  expressions  : 

(a)  a  +  a^  =  a  -{-  a^. 

(b)  alb'ip>a\lb>  1. 

(c)  a^  ^L  a  -\-  a^  —  a,  .' .  a"^  <.  a  -\-  aP'. 

.(d)  •••  a  =  2,  .'.  a"  =  4.,  a^  =  S,  a^  =  16,  •  •  •. 
(e)  a^  -{-  a^  =^  a^,  and  a^  +  a^  ^  a^,  if  a  is  positive. 

3.  Show  that  the  following  are  equal  when  a  =  2  and 
b  =  8.  That  is,  substitute  2  for  a  and  3  for  b  in  each 
member. 

(a)  (a  +  by==a^  +  2ab  +  b\ 

,(b)  (b-ay  =  b^-2ba  +  a^ 

(c)  (b^-a^)/(b-a)=b  +  a. 

(d)  (a  +  b)  (a^  -  ab  +  b^)  =  a^ -\-  b\ 

(e)  {b""  -  a"")  /  (b  -  a)  =  b''  ■}-  ba  +  a\ 

(f )  {a  +  by  =  a^  +  8  a%  +  3  ah^  +  b\ 


INTRODUCTION  TO   ALGEBRA.  25 


V.     PROPOSITIONS   OF  ALGEBRA. 

35.  A  proposition  is  a  statement  of  either  a  truth  to  be 
demonstrated  or  something  to  be  done. 

E.g.,  algebra  investigates  this  proposition  :  The  product  of  a"*  and 
Qn  ig  dm  +  n_  j^  also  considers  sucli  statements  as  this  :  Required  the 
product  of  a  -\-  b  and  a  —  b. 

36.  Propositions  are  divided  into  two  classes,  theorems 
and  problems. 

A  theorem  is  a  statement  of  a  truth  to  be  demonstrated. 
E.g.,  The  product  of  a"*  and  a"  is  «"»  +  ". 

A  problem  is  a  statement  of  something  to  be  done. 
E.g.,  Required  the  product  of  a  +  b  and  a  —  b. 

A  corollary  is  a  proposition  so  connected  with  another  as 
not  to  require  separate  treatment. 

The  proof  is  usually  substantially  included  in  that  of  the  proposition 
with  which  it  is  connected. 

REVIEW   EXERCISES.    IX. 

1.  What  is  the  degree  of  the  expression  3  ax^y^  ?  What 
is  its  degree  in  ic  ?    in  y?    in  ic  and  y?   in  z? 

2.  Distinguish  between  coefficient  and  exponent.  What 
is  the  coefficient  of  x  in  the  expression  -  ?    the  exponent  ? 

3.  What  is  the  meaning  of  the  expression  ab  ?  of  26  ? 
of  a  I  ?    of  2f  ?     What  is  the  value  oi  ab  it  a  =  2,  b  =  6? 

ofafiia  =  2,x  =  3,y  =  4:? 

4.  What  is  meant  by  the  etymology  of  a  word  ?  What 
is  the  etymological  meaning  of  binomial?  of  trinomial? 
of  monomial  ?  of  aggregation  ?  of  theorem  ?  (See  Table 
of  Etymologies.) 


26  ELEMENTS  OF  ALGEBRA. 

5.  Show  that  if  a  =  7  and  h  =  5, 

(a)  (a  +  b)(a-b)  =  a^-P. 

(b)  (a-by  =  a''  +  P-2ab. 

(c)  (a  -  by  =  a^  -  b^  -  3ab(a  -  b). 

6.  Show  that  a  a  =  3,  b  =  2,  c  =  1, 

(a)  (a-]-by-c'^  =  (a  +  b  +  c)(a  +  b  -  c). 

(b)  la  +  b  +  cy  =  a^  +  b^-{-c^  +  2ab-\-2bc  +  2  ca. 

7.  What  meaning  has  the  number  "  minus  2  "  to  you  ? 

8.  What  is  the  value  of  8^  ?   of  9^  of  16^  •  32^  --  16'  ? 

9.  Show  by  substitution  that  1  is  a  root  of  the  equation 
in  ex.  10. 

10.  How  many  terms  in  the  equation  2x^-\-3x  — 4^  =  1? 
How  many  members  ? 

11.  Draw  a  diagram  illustrating  the  fact  that  the  abso- 
lute value  of  the  difference  between  —  5  and  10  is  15. 

12.  What  is  the  degree  of  the  polynomial  x^  +  3x^y^  + 
3  xy^  -\-  5y  +  6  ?     What  is  the  degree  in  a?  ?    my?   in  z? 

13.  Write  the  following  in  algebraic  language :  The  sum 
of  the  square  of  a  number,  3  times  the  number,  and  5,  is 
equal  to  9. 

14.  Represent  algebraically  the  sum  of  the  cube  of  a 
number,  5  times  the  square  of  the  number,  and  6,  less  half 
the  number. 

15.  What  is  meant  by  solving  an  equation  ?  by  a  root  of 
an  equation  ?  by  checking  a  solution  ?  Illustrate  with  the 
equation  ic  —  2  =  0. 

16.  What  is  the  number  from  which  if  5%  be  taken,  and 
10%  from  the  remainder,  and  20%  from  that  remainder, 
the  result  is  41.04? 

17.  Write  ou^  thi-ee  problems  which  you  can  now  solve, 
but  which  you  could  not  solve  when  you  began  to  study 
algebra. 


CHAPTER   II. 

ADDITION   AND    SUBTRACTION. 

I.     ADDITION. 

37.  In  elementary  arithmetic  the  word  number  includes 
only  positive  integers  and  fractions,  or  at  most  a  few  indi- 
cated roots  like  V2,  VS,  •  •  • .  Hence,  the  word  sum,  as  there 
used,  applies  only  to  the  result  of  adding  two  positive  num- 
bers. 

In  algebra  the  word  sum  has  a  broader  meaning,  and 
includes  the  results  of  adding  negative  numbers  and  num- 
bers some  of  which  are  positive  and  others  negative. 

E.g.^  consider  the  combined  weight  of  these  three  articles:  a  2-11). 
weight,  a  4-lb.  weight,  and  a  balloon  which  weighs  —  5  lbs.  (i.e.,  pulls 
upward  with  a  force  of  5  lbs.).  Together  they  would  evidently  weigh 
1  lb.     Hence  1  lb.  is  said  to  be  the  sum  of  2  lbs.,  4  lbs.,  and  —  5  lbs. 

So  the  result  of  adding  a  debt  of  $100  to  a  capital  of  $300  is  a  capital 
of  $200  ;  hence,  $200  is  said  to  be  the  sum  of  $300  and  -  $100. 

38.  In  this  broader  view  of  addition  two  cases  evidently 
arise : 

1.  Numbers  with  like  signs. 

2  lbs.  +  3  lbs.  =  5  lbs. 

A  balloon  pulling  up  5  lbs.  and  one  pulling  up  8  lbs.  together  pull 
up  13  lbs.,  or  (-  5  lbs.)  +  (-  8  lbs.)  =  -  13  lbs. 

2.  Numbers  with  unlike  signs. 

A  balloon  pulling  up  5  lbs.  and  a  weight  of  2  lbs.  together  pull  up 
3  lbs.,  or  -  5  lbs.  +  2  lbs.  =  -  3  lbs. 

27 


28  ELEMENTS   OF  ALGEBRA. 

39.  From  considerations  like  these  we  are  led  to  define 
the  sum  of  two  algebraic  numbers  as  follows : 

1.  If  two  numbers  have  the  same  sign,  their  algebraic  sum 
is  the  sum  of  their  absolute  values,  preceded  by  their  common 
sign. 

Thus,  to  add  —  3  and  —  2  means  to  add  3  and  2  and  to  place  the 
sign  —  before  the  result. 

2.  If  they  have  not  the  saTue  sign^  their  algebraic  sum  is 
the  difference  of  their  absolute  values,  preceded  by  the  sign 
of  the  one  which  has  the  greater  absolute  value. 

Thus,  to  add  —  3  and  2  means  to  find  the  difference  between  3  and 
2  and  to  place  the  sign  —  before  the  result,  since  I  —  3  I  >  I  2  I. 

3.  In  the  special  case  where  the  two  numbers  have  the 
same  absolute  value  (i.e.,  where  they  are  equal  and  of  oppo- 
site signs),  the  sum  is  zero. 

E.g.,2  +  {-2)  =  0. 

4.  If  one  of  two  numbers  is  zero,  their  algebraic  sum  is 
the  other  number. 

Thus,  -3  +  0  means  -  3. 

40.  The  algebraic  sum  of  several  numbers  is  defined  as 
the  sum  of  the  first  two  plus  the  third,  that  sum  plus  the 
fourth,  ■  •  • . 

Thus,  a  +  h  +  c  +  d  means  a  +  &  with  c  added,  and  that  sum  with 
d  added.     I.e.,  a  +  6  +  c  +  d  means  [(a  +  6)  +  c]  +  d. 

EXERCISES.    X. 

1.  Find  the  sum  of  -  20,  +  3,  -  47,  +  80. 

2.  Also  of  +  2,  -  3,  +  5,  -  4,  +  9,  -  3,  -  6. 

3.  Also  of  2  x%  5  x2,  -  6  x^  Sx^? 

4.  Also  of  127  mn,  62  mn,  —  93  mn,  -- 17  mn? 


ADDITION   AND   SUBTRACTION.  29 

5.  $50 +  17 +(-$21)  + (-130)=? 

6.  5  +  219  +  (-  376)  +  (-  40)  +  10  +  (-  37)  =  ? 

7.  (-  7)  +  4  +  (-  2)  +  18  +  13  +  (-  20)  +  (-  6)  =  ? 

8.  Sa-{-(-2a)  +  (-5a)  +  8a  +  6a  +  (-10a)  =  ? 

9.  What  is  the  sum  oi  3  a,  5  a,  —6  a,  Sa,  10  a,  —  3  a, 
-17a? 

10.  12  x^^  +  4  ccV  +  (-  16  x^y)  +  (-  3  xhj)  +  10  xhj 
=  (?)x'y? 

11.  5  lbs.  +  55  lbs.  +  (-  40  lbs.)  +  (-  27  lbs.)  +  121  lbs. 
+  (-  19  lbs.)  +  (-  5  lbs.)  =  (?)  lbs  ? 

12.  What  is  the  combined  weight  of  two  balloons  weigh- 
ing, respectively,  —  10  lbs.  and  —  18  lbs.,  and  thi-ee  pieces 
of  iron  weighing,  respectively,  6  lbs.,  12  lbs.,  and  14  lbs.  ? 

13.  On  seven  consecutive  midnights  in  January,  in  Mon- 
treal, the  temperature  was  30°,  18°,  10°,  4°,  0°,  -  7°,  -  20°. 
What  was  the  average  midnight  temperature  for  the  week  ? 

14.  What  is  the  combined  weight,  under  water,  of  a  piece 
of  cork  weighing  —  2  oz.,  a  stone  weighing  3  lbs.,  a  piece  of 
wood  weighing  —  1  lb.  3  oz.,  and  a  piece  of  iron  weighing 
5  lbs.  ? 

15.  A  merchant  finds  that  he  has  cash  in  bank  $575.50, 
stock  worth  $4875,  due  from  customers  $1121.50,  that  he 
owes  a  note  and  interest  amounting  to  $350.25  and  bills 
amounting  to  $827,  and  that  he  owns  a  bond  and  mortgage 
of  $1000.  Express  his  capital  as  the  sum  of  these  various 
items  with  their  proper  signs. 

16.  A  ship  sailing  up  a  river  would  go  at  the  rate  of  15 
miles  an  hour  if  it  were  not  for  the  current ;  the  current 
averages  5  miles  an  hour  for  the  first  3  hours  of  the  ship's 
progress,  and  4  miles  an  hour  for  the  next  2  hours.  How 
far  has  the  ship  gone  at  the  end  of  5  hours  ?  Express  this 
as  the  sum  of  several  algebraic  numbers. 


2a+     6  -  3c 

46+     c 

-  (5  a  -     6  +     c 

-4a  +  46  -     c 

4  +  i-15=- 
2+    5  = 
-  12  -  i  +    5  =  - 

lOi 

7 

30  ELEMENTS   OF  ALGEBRA. 

41.  To  add  several  literal  expressions,  called  the  addends, 
is  to  find  a  single  expression  called  the  sum,  such  that  what- 
ever values  are  substituted  for  the  letters  the  value  of  the 
sum  shall  equal  the  sum  of  the  values  of  the  addends. 

E.g.,  the  sum  of  a,  2  a,  7  a,  —  4  a,  is  6 a ;  for  suppose  1  is  substi- 
tuted for  a,  we  have  1+2  +  7  —  4,  which  is  6  ;  and  if  2  is  substituted 
for  a  we  have  2  +  4  +  14  —  8,  which  is  12, 
and  so  for  any  other  values. 

Similarly,  the  sum  of  the  addends  in  the 
annexed  problem  is  —  4  a  +  4  6  —  c  ;  for  if 
a  =  2,  6  =  i,  c  =  5,  we  have  —  10^  +  7  —  7i 
=  —  11,  and  similarly  for  any  other  values 
of  a,  6,  c. 

Since   these   values    are    entirely 
arbitraiy,    they    are    usually    called      _   8  +  2-   5=-ll 
arbitrary  values. 

42.  Hence,  it  appears  that  to  add  like  terms  is  to  add  the 
coefficients,  and  to  add  polynomials  is  to  add  their  like 
terms,  the  literal  parts  being  properly  inserted  in  the  sum. 

The  sum  is  supposed  to  be  simplified  as  much  as  possible.  Thus, 
the  sum  of  4  a  —  6  and  6  +  a  is  5  a,  not  4  a  +  a. 

EXERCISES.    XI. 

1.  Add  ^x'^^2xy-\-4.y%  ^x^ -?>xy  -'ly'',  ^x''  +  xy. 

2.  Add  6Vm -{- X,  dVm  —  X  —  3y,  SVm —  2y,Siiid  8x. 
Check  the  work  by  letting  m  =  4:,  x  =  1,  y  =  1. 

3.  Add  2a  +  Sb  —  c,  — 4c,  7a,  —6b-\-Sc,  and  —  a-\-b 
—  c.     Check  the  work  by  letting  a  =  1,  b  =  1,  c  =  1. 

4.  Add  17 X  —  9y,  Sz  +  14:X,  y  —  3x,  x  —  17 z,  and  x  — 
3y  -\-  4:Z.     Check  the  work  by  letting  x  =  1,  y  =  2,  z  =  3. 

5.  Add  16  m  -^  3  n  —  p,  p  -\-  4:  q,  —  q  -\-  7  m  —  3  n,  n  —  q, 
and  3n  -\-  2 p.  Check  the  work  by  letting  m  =  l,  n  =  l, 
p  z=z  2,  q  =  4:,  01  hy  assigning  any  other  arbitrary  values. 


ADDITION   AND   SUBTRACTION.  31 

II.     SUBTRACTION. 

43.  Subtraction  is  the  operation  which  has  for  its  object, 
given  the  sum  of  two  expressions  and  one  of  them,  to  find 
the  other. 

The  given  sum  is  called  the  minuend,  the  given  addend  is 
called  the  subtrahend,  and  the  addend  to  be  found  is  called 
the  difference  or  the  remainder. 

That  is,  the  difference  is  the  number  which  added  to  the 
subtrahend  produces  the  minuend.     In  other  words, 

difference  +  subtrahend  =  minuend. 

E.g.,         •.•4  +  5  =9,  .-.      4  =  9-5; 

v4  +  (-3)       =1,  .-.       4  =  1 -(-3); 

v4  +  (-5)       =-1,  .-.       4=-l-(-5); 

...  -  4  +  {-  3)  =  -  7,  .-.  _  4  =  -  7  -  (-  3). 

These  results  are  illustrated  as  follows:  the  difference 
between  the  temperature  of  9°  and  that  of  5°  is  4°;  that 
between  1°  and  -3°  (i.e.,  1°  above  0  and  3°  below  0)  is  4°; 
that  between  -  1°  and  —  5°  {i.e.,  1°  below  0  and  5°  below  0) 
is  4° ;  that  between  —  7°  and  —  3°  is  —  4°,  that  is,  the  mer- 
cury must  fall  4°  from  -  3°  to  reach  -  7°. 

We  may,  therefore,  think  of  subtraction  as  the  inverse  of 
addition,  or  the  process  which  undoes  addition. 

Example.     What  is  the  remainder  after  subtracting 
3  a2  +  4  a6  -  5  62  from  4  a^  -  6  a6  +  2  62  ? 

What  term  added  to  3  a2  makes  .9      k    ».  ,  o  7,9 

4  a''  —  5  a6  +  2  6-^ 
4a2?     Evidently  a2.  3a2  +  4a6-562 

What  term  added  to  4  ah  makes  — ; — TnrvWTi 

a^  —  9ao  +  1  0^ 
-5 ah?    Evidently  -  9 a6 ;  for  the 

addition  of  —  4  a6  makes  0,  and  the  further  addition  of  —  5  a6  makes 

—  5a6. 

Similarly,  7  62  is  the  term  v^^hich  added  to  —  5  62  makes  2  62. 

.-.  4  a2  -  5  a6  +  2  62  -  (3  a2  +  4  a6  -  5  62)  =  a2  -  9  a6  +  7  62. 


32  ELEMENTS   OF  ALGEBRA. 

Check.   Let  a  =  1,  6  =  2.   Then 
4a2-5a6  +  262  4  _  lo  +    8  =      2 

3a2  +  4ab-5b2  3+    8- 20  =-9 

a2  -  9  a&  +  7  62  1  -  18  +  28  =    11 

Since  this  is  an  identity,  it  is  true  for  any  values  of  a  and  h.  Hence, 
the  work  may  be  checked  by  letting  a  =  1,  6  =  2.  The  minuend  then 
becomes  2,  and  the  subtrahend  —  9,  and  the  remainder  11,  which  is 
2 -(-9). 

44.    Theorem.    The  subtraction  of  a  negative  number  should 
be  interpreted  as  the  addition  of  its  absolute  value. 
Given         a  and  —  b. 
To  prove    that  a  —{—b)  equals  a  plus  the  absolute  value 

of  —  5  ;   that  is, 

that      a—(^—b)=a-\-\  —  b\oYa  +  b. 

Proof.    1.  a  —  {—b)  must  be  such  a  number  that 

a-{-b)  +  {-b)=  a.      Def.  of  subt.,  §  43 
2.  Adding  b  to  both  members,  and  remembering 
tliat  (_  J)  +  ^  =  0,  §  39,  3 

and  that  a -\- 0  =  a,  §  39,  4 

we  have       a  —  (—  ^)  =  <x  +  &, 
which  we  were  to  prove. 

Corollary..  *.•  a  —  {—b)  =  a-\-b,  .' .  0  —  (—  b)  =  b.  This 
is  usually  expressed  by  the  phrase,  Minus  a  minus  is  plus. 

EXERCISES.    XII. 

1.  From  3a  —  4:b  -\-c  subtract  2a  —  5b  —  c.  Check  the 
work  by  letting  a  =  3,  b  =  1,  c  =  2,  or  by  assigning  any 
other  arbitrary  values ;  also  by  adding  the  remainder  and 
subtrahend. 

2.  Prom  3a  —  5cc  +  fm  subtract  4 a  —  m.  Check  by 
letting  a  =  5,  x  =  1,  m  =  3,   or    by   assigning  any   other 


A 


ADDITION  AND   SUBTRACTION.  33 

arbitrary  values ;    also  by  adding  the  remainder  and  sub- 
trahend. 

3.  From  13x  +  y  —  3z  subtract  5 x  —  7 1/  +  z.     Check 
as  in  exs.  1,  2. 

4.  From    Ta""  -\-3ab  -  6b^   subtract    a''  +  3  ab  -  2  b\ 
Check  as  in  exs.  1,  2. 

5.  What  expression  added  to  2  x^  —  3  x?/  —  15  y^  makes 
-7x'^-3xij  +  z?     Check. 

6.  Perform  the  following  subtractions,  checking  each  as 
in  exs.  1,  2. 

(a)  3ic3-7a;2  +  2ic-13  (b)   l.Sjy^  -  2p^r  +  O.Sr^ 

(c)     2a^-3ab-{-      b  -       c 
V  17  ai  -13b -{-12c 


(d)  2a^-3a''b-\-2  a^b^  -  3  a%^  +  15  ab''  -  b' 
6a^ -f-  3  a^b^  -     a%'' -j-J^ 

(e)          18^2^,^  + 4a  -3Z>«  (f)  2xy-y''-^3x' 

1  a^ -2a  +  4^>^-     6^  x''-\-y''-3xy 

7.  What  is  the  difference  between  the  capital  of  a  man 
who  has  a  stock  of  goods  worth  $5000,  $750  in  the  bank, 
and  owes  $1000  on  a  mortgage,  and  that  of  one  who  has 
a  stock  of  goods  worth  $6000,  has  overdrawn  his  bank 
account  $275,  and  owns  a  $500  mortgage  ? 

8.  If  P  =  a2  +  2  a6  +  52^  g  =  2a2  +  aZ»  -f-  6^,  and  B  = 
—  4:ab  —  7  b^,  find  the  values  of  the  following  expressions. 
Check  in  each  case  by  assigning  arbitrary  values  to  a  and  b. 

(a)  F-Q.  (b)  P  -  P.  (c)   Q-E. 

(d)   Q-P.  .-    -        ^       -         .„.    _       _ 

(g)  F-Q-E. 


(b)  F-E.  (c)   Q-E. 

(e)  F-{-Q-E.       {i)  F  +  E-Q. 
(h)   E-Q-F.        (i)   Q-F-E. 


34  ELEMENTS   OF  ALGEBRA. 

45.  Detached  coefficients.  Additions  and  subtractions  may 
evidently  be  performed  without  the  labor  of  writing  down 
all  of  the  letters.  Since  the  coefficients  of  like  terms  are 
added,  these  coefficients  may  be  detached  and  added  sepa- 
rately, the  coefficients  of  like  terms  being  placed  under  one 
another.  Missing  terms  are  indicated  by  zeros. 
Thus,  the  second  of  the  following  additions  is  the  simpler : 
(1)  (2)  Check. 

a2  +  2a6+&2  1  +  2  +  I         =      4 


_3a2-    a6+    h^ 

-3-1+1 

=  -3 

4a2-3a&-362 

4-3-3 

=  -2 

2a2_2a6-    &2 

2-2-1 

=  -  1 

2a2. 

-2 ah-  62. 

Since,  if  the  arbitrary  value  1  is  assigned  to  each  letter,  the  value 
of  each  term  is  its  numerical  coefficient,  the  check  requires  merely  the 
addition  of  the  coeflBcients. 

EXERCISES.    XIII. 

Perform  the  following  operations  by  using  detached 
coefficients,  checking  the  results  by  the  above  method. 

1.  Add  a%  +  a%^  -  4  a^^  3  a«^>  -  h\  -  a%^  +  h\  4  ab^. 

2.  Add  5  a;*  —  2  x^if'  +  y^,  x^y  +  xy^,  x^  —  xy^,  —  x^y  +  y\ 

3.  Add  x^  —  x^y  +  xy^  —  y^,  2x^  +  3  x^y  —  4  xy^  +  y^, 
x^  —  y^. 

4.  Add  p^  4-  Sp^  +  4j9  -  6,  -y  _  2j9  +  1,  p3  _  ^^  3^ 
+  2p  +  3. 

5.  From  a''  +  2ab  +  b^  subtract  a^  -  2  ab  +  b'i 

6.  From  x^  +  x^y  +  xy^  +  y^  subtract  x^  —  x^y  +  xy^  —  y^ 

7.  Given    P  =  x''  +  Sx'y  +  3xy'  +  f,    Q=  -  3x'y  + 
3  xy^  —  3y^,E  =  x^  —  y^,  find  by  using  detached  coefficients 
the  values  of  the  following,  checking  as  above : 
(a)P-g.     (h)Q-R.     (G)R-P.  (d)Q-F. 
(e)Ii-Q.     (i)F-R.     (g)F-^Q  +  B.     (h)P-{.Q-E. 


ADDITION   AND    SUBTRACTION.  35 

III.     SYMBOLS    OF  AGGREGATION. 

4^.  Symbols  of  aggregation,  preceded  by  the  symbols  + 
and  — ,  may  be  removed  by  considering  the  principles  of 
addition  and  subtraction  already  learned. 

Since  a  -{-(b  —  c)  =  a  -{-  b  —  c, 

and  a  —  (b  —  c)  =  a  —  b  -\-  c, 

therefore,  a  symbol  of  aggregation  preceded  by  -\-  may  be 
neglected;  if  preceded  by  —  it  may  be  removed  by  changing 
the  sig7i  of  each  term  within. 

E.g.,  2a  +  (3&-c  +  a)  =  2a-\-Sb-  c  +  a  =  Sa -\- Sb  -  c. 
2a-(3  6-c  +  a)  =  2a-3&  +  c-a  =  a-36  +  c. 

For  the  same  reasons,  any  terms  of  a  polynomial  may  be 
enclosed  in  a  symbol  of  aggregation  preceded  by  +  ;  also  in 
a  symbol  of  aggregation  preceded  by  —  provided  the  sign 
of  each  term,  within  is  changed. 

E.g.  ,a  +  &  —  c  +  (Z  =  a  +  (6  —  c  +  d)  =  a  +  &  —  (c  —  d). 

The  word  term  now  takes  on  a  broader  meaning  than  that 
given  in  §  3.  E.g.,  in  the  expression  a  —  b(G  —  d),  b(c  —  d) 
is  often  considered  as  a  term.  So  in  general,  where  no  con- 
fusion will  arise,  polynomials  enclosed  in  symbols  of  aggre- 
gation, with  or  without  coefficients,  are  often  called  terms. 

E.g.,  (a  —  6)x2  +  (a  +  6)x  -I-  (a^  —  h^)  may  be  considered  as  a  tri- 
nomial. 

EXERCISES.    XIV. 

Kemove  the  symbols  of  aggregation  in  the  following : 

1.  p^  +  2pq  +  q'-(q'-p^). 

2.  a''-3b''-h(2a^  +  7b''-c^). 

3.  a^  -  (3  a'^b  Jra^-b^)-b^  +  3  a%. 

4.  2x'^-3xy  +  y^-{2x''  +  3xy-y^). 

5.  5m^  -  (3  m^  -f-  1)  -  (4  7/2.^  +  m^  -  3)  +  (m^  +  1). 


36  ELEMENTS   OF   ALGEBRA. 

47.  Several  symbols  of  aggregation,  one  within  another, 
may  be  removed  by  keeping  in  mind  the  principles  already 
mentioned. 

The  order  in  which  these  symbols  are  removed  cannot 
affect  the  result,  but  the  simplest  plan  will  be  discovered 
by  considering  the  following  solution. 

Simplify  a  —  \^a  -\-  b  —  (c  —  d  —  e)  -\-  c"],  (1)  beginning 
with  the  inner  symbol,  (2)  beginning  with  the  outer  symbol. 

(1)  (2) 


1.  a  —  [a  +  h  —  (c  —d  —  e)  +  c]     1.      a—  [a  +  b  —  {c  —d  —  e)  -\-  c] 

2.  =a  —  [a  +  5  —  (c  —  d  +  e)  +  c]     2.  =a  —   a  —  b  +  (c—d  —  e)—c 

3.  =a  —  [a-\-h—  c  +  d  —  e  +  c]     3.  =a  —  a  —  h+  c—d  —  e  — c 

4.  ^a  —  a-b+c  —  d-\-e— c      4.  =a  —   a  —  6+  c  —d  +  e  —  c 

5.  =  —  b         —  d  -\-  e  6.  =  —b         —d  +  e 
How  many  changes  of  signs  were  made  throughout  solution  (1)  ? 

how  many  in  solution  (2)  ?     Hence,  which  solution  is  the  better  ? 

From  the  second  step  of  solution  (2)  could  you  have  written  down 
step  5  at  once  ?  Could  you  have  done  this  from  step  2  of  solution  (1)  ? 
On  this  account  which  is  the  better  solution  ? 

From  the  above  solution  it  appears  that  it  is  better  to 
rem,ove  the  outer  parentheses  first.  A  little  practice  will 
enable  the  student  to  remove  them  all  at  sight  if  this  plan 
is  followed. 

EXERCISES.    XV. 

Remove  the  symbols  of  aggregation  in  the  following 
expressions,  uniting  like  terms  in  each  result. 

1.  -  [a^  -{2ab-b''-  w")  +  b'^y 

2.  4.a''-\5b''  +  a-[^&a'-3a-{b''-a)-\\, 

3.  a^x  —  [_ax^  -\-  a^  —  (a^x  —  a^)  +  x^~\  —  ax"^  +  x^. 


4.  10  m^  +  5  mn  —  [6  m^  +  ti^  —  (2  mn  —  m^  -\-  n'^)']  —  n^. 

5.  —(—(—(•••  —  (—  1)  •  •  •))),  an  even  number  of  sets 
of  parentheses ;    an  odd  number  of  sets. 


I 


ADDITION  AND   SUBTRACTION.  37 

IV.     FUNDAMENTAL  LAWS. 

48.  The  following  laws  have  thus  far  been  assumed : 

I.  That  a  +  b  =  b  -\-  a,  a  and  b  being  positive  or  negative 
integers,  just  as  in  arithmetic  3  +  4  =  4  +  3.  This  is 
called  the  Commutative  Law  of  Addition,  because  the  order 
of  the  addends  is  changed  (Latin  com,  intensive,  +  mutare, 
to  change). 

II.  That  a-\-b-\-c^a-{-(b-\-c),  the  letters  represent- 
ing positive  or  negative  integers,  or  both,  just  as  in  arith- 
metic 3  +  4  +  5  =  3  +  9.  This  is  called  the  Associative 
Law  of  Addition,  because  b  and  c  are  associated  in  a  group. 

III.  That  ab  =  ba,  a  and  b  being  positive  integers,  just 
as  in  arithmetic  2  •  3  =  3  •  2.  This  is  called  the  Commuta- 
tive Law  of  Multiplication. 

That  these  laws  are  valid  for  the  kinds  of  numbers  indi- 
cated will  now  be  proved,  although  the  proof  may  be 
omitted  by  beginners  if  desired. 

49.  I.  The  Commutative  Law  of  Addition". 

1.  If  3  marbles  lie  on  a  table,  and  4  more  are  placed  with 
them,  the  result  is  indicated  by  the  symbols  3+4. 

2.  If  the  original  3  marbles  be  removed,  4  will  remain ; 
and  if  the  3  be  then  replaced,  the  result  will  be  indicated 
by  the  symbols  4  +  3. 

3.  But  the  number  of  marbles  has  not  been  changed. 

.-.3  +  4  =  4  +  3. 

4.  But  this  proof  is  independent  of  the  particular  nmn- 
bers  3  and  4,  and  hence,  a  and  b  being  any  positive  integers, 

a  -\-  b  ^^  b  -{-  a. 

5.  The  proof  is  evidently  substantially  the  same  for  sev- 
eral groups.     Hence, 

a  +  6  +  c  +  -"  =  a  +  c  +  6H —  •  =  b  -{-  c  ->r  a  -\ ,  etc. 


38  ELEMENTS   OF  ALGEBRA. 

6.  And  since,  if  some  of  the  terms  are  negative,  we  deal 
with  their  absolute  values,  adding  or  subtracting  as  indi- 
cated, and  prefix  the  proper  sign  to  the  result,  therefore 
the  above  proof  is  sufficiently  general. 

I.e.,  a  +  b  —  c  =  a  —  c  +  b,  because  in  any  case  we  are  to  take  the 
difference  between  the  absolute  values  of  a  +  6  and  c,  and  prefix  the 
proper  sign. 

50.  II.  The  Associative  Law  of  Addition. 

To  prove  that  a  -\-  b  +  c  =  a  -\-  (b  +  c),  the  letters  repre- 
senting positive  or  negative  integers,  or  both. 

1.  '.•c  +  b  +  a  =  (c  +  b)  +  a.  Def.,  §  40 

2.  =  a+(c  +  b).  ■      Com.  law,  §  49 

3.  .-.  a  +  b  +  G=  a  +(b  -\-c).  Com.  law,  §  49 
The  proof  is  evidently  similar,  however  many  terms  are 

involved  or  however  the  grouping  is  made. 

51.  III.  The  Commutative  Law  of  Multiplication. 
To  prove  that  ab  =  ba,  the  letters  representing  only  posi- 
tive integers. 

*****...       a  in  a  row 


b  rows. 

1.  Suppose  a  collection  of  objects  arranged  in  b  rows,  a 
in  a  row,  or,  what  is  the  same  thing,  in  a  columns,  6  in  a 
column. 

2.  •.'  there  are  b  in  one  colmnn,  in  a  columns  there  are 
ab  objects. 

3.  *.•  there  are  a  in  one  row,  in  b  rows  there  are  ba  objects. 

4.  But  the  collection  being  the  same,  ab  =  ba. 


ADDITION  AND   SUBTRACTION.  39 

REVIEW   EXERCISES.    XVI. 

1.  Distinguish  between  an  equation  and  an  identity,  illus- 
trating each. 

2.  Show  that  |2  —  3|  =  |3  —  2|,  and  state  a  proposition 
covering  such  cases. 

3.  What  is  the  etymological  meaning  of  coefficient  ?  of 
subtraction  ?    of  literal  ?    of  minuend  ? 

4.  Why  is  not  the  arithmetic  definition  of  sum  sufficient 
for  algebra  ?     What  do  you  mean  by  sum  in  algebra  ? 

5.  What  is  the  advantage  in  using  detached  coefficients 
in  addition  ?     Make  up  an  example  illustrating  this. 

6.  What  is  the  number  which  added  to  —  5  equals  0  ? 
equals  2  ?  Hence,  what  is  the  difference  0  —  (—  5)  ? 
2-(-5)? 

7.  Eemove  the  symbols  of  aggregation  in  the  following 
expressions.  By  beginning  at  the  outside  you  can  usually 
write  the  result  at  sight,  except  for  simplifying. 


(a)  [x+{x  +  ^j  -{-x-y)-  x']. 

(b)  a —  \a —\_a  —  a —  {b  —  a)~\\. 

(c)  Z  a  -[b  +  c  -  (a  -  b)  -{-  a']—  h. 

(d)  x''-[2x^  +  y'-(x^  -  2/2  -  ^?^=^0  +  2/']-  2/'. 

8.  Enclose  any  two  terms  (after  the  first)  in  parenthe- 
ses: 

(a)  «;2_^2_2c2_3^»c.        (b)  Zp''-4.pq-2q''  +  r\ 

(c)  4  ic^  -  2  x2  -  7  ic  +  1.        (d)   m'^  —  m^  +  m^  -  m  +  1. 

9.  What  is  meant  by  the  Commutative  Law  of  Addi- 
tion ?  Have  you  proved  it  for  all  kinds  of  numbers  ?  If 
not,  name  a  kind  for  which  it  has  not  yet  been  proved  by 
you.     Similarly  for  the  Associative  Law  of  Addition. 


CHAPTER   III. 

MULTIPLICATION. 

I.     DEFINITIONS   AND   FUNDAMENTAL   LAWS. 

52.  Multiplication  originally  had  reference  to  positive 
integers  and  was  a  short  form  of  addition.  It  was,  for 
this  case,  defined  as  the  operation  of  taking  a  number 
called  the  multiplicand  as  many  times  as  an  addend  as  there 
are  units  in  an  abstract  number  called  the  multiplier,  the 
result  being  called  the  product. 

E.g.,  in  this  limited  sense,  to  multiply  .$2  by  3  is  to  take  $2  3  times, 
thus,  3  X  $2  =  $2  +  $2  +  $2  =  $6. 

But  as  mathematics  progressed  it  became  necessary  to 
multiply  by  simple  fractions,  and  hence  to  enlarge  the  defi- 
nition to  include  this  case. 

By  the  primitive  meaning  of  the  word  times  it  is  impossible  to  take 

2  X  $2 

$2  f  o/  a  time.     But  the  product  of  $2  by  f  may  be  defined  as 

o 

So  the  product  of  c  by  -  may  be  defined  as  the  product 

of  a  and  c,  divided  by  b,  c  being  either  integral  or  fractional. 
As  mathematics  further  progressed  it  became  necessary 
to  multiply  by  negative  numbers,  and  hence  to  enlarge  the 
definition  to  include  this  case.  The  natural  definition  will 
appear  from  a  simple  illustration. 

Suppose  5  men  move  into  a  town,  each  paying  $  1  a  week  in  taxes. 
They  are  worth  5  x  $l  =  $5a  week  to  the  town. 

Suppose  5  such  men  move  out.  This  may  be  represented  by  saying 
that  the  town  gains  —  5  men,  or,  in  money,  —  $5. 

40 


MULTIPLICATION.  41 

Suppose  5  vagrants  move  in,  each  being  a  charge  of  $1  a  week. 
They  are  worth  5x(— $1)  =  —  -fSa  week  to  the  town. 

Suppose  5  such  vagrants  move  out.     This  may  be  represented  by 
saying  that  the  town  gains  —  5  vagrants,  or,  in  money,  $  5. 
Hence,  it  is  reasonable  to  say  that 

$  1  multiplied  by      5  =      $  5,  for  the  first      case ; 
$1         "         "      —5  =—$5,      "       second    " 
-.fl         "         "  5=  -15,      "       third       " 

-$1         "         "      -5=      $5,      "        fourth    " 

53.  From  such  considerations  multiplication  by  a  negative 
number  is  defined  as  multiplication  by  the  absolute  value  of 
the  multiplier,  the  sign  of  the  product  being  changed. 

E.g.,  allowing  the  word  times  to  indicate  multiplication  in  general, 
-  2  times  3  means  -  ( 1  -  2  I  x  3),  or  -  (2  x  3),  or  -  6  ; 
_2  "  -3  "  _  [I -21  X  (-3)]  "  -[2x(-3)]  "  -(-6), 
or  +  6. 

54.  General  definition  of  multiplication.  The  above  partial 
definitions  may  now  be  put  into  one  general  definition : 

To  multiply  a  number  (the  multiplicand)  by  an  abstract 
numher  (the  multiplier)  is  to  do  to  the  former  what  is  done 
to  unity  to  obtain  the  latter. 

The  result  of  multiplication  is  called  the  product,  and  the 
product  of  two  abstract  numbers  is  called  a  multiple  of  either. 

E.g..,  consider  the  meaning  of  3  x  $2.  Since  3  =  1  +  1+1,  there- 
fore, 3  X  $2  means  $2  +  $2  +  $2  =  $6. 

Consider  also  f  x  f .  Since  |  =  (1  +  1)  -4-  3,  therefore,  f  x  f  means 
(f  +  f)  -  3,  or  -V-  ^  3,  or  if 

Consider  also  (—2)  x  (—3).  Since  —  2  =  —  (1  +  1),  therefore, 
(-  2)  X  (-  3)  means  -  [(  -  3)  +  (-  3)],  or  -  (-  6),  or  6. 

55.  The  expression  a  •  0  is  defined  to  mean  0. 

This  is  the  natural  definition,  because  2x0  must  mean  0  +  0. 

And  since  it  will  be  shown  that  the  order  of  factors 
can  generally  be  changed  without  altering  the  product,  the 
product  0-Si  is  defined  to  be  the  same  as  a  •  0,  or  0. 


42  ELEMENTS   OF  ALGEBRA. 

56.  The  product  of  three  abstract  numbers  is  defined  to  be 
the  product  of  the  second  and  third  multiplied  by  the  fii-st. 

J.e.,  abc  means  a{bc),  the  product  of  6  and  c  multiplied  by  a. 

The  product  of  four  or  more  abstract  numbers  may  be  understood 
from  the  above  definition  for  three,  ^.g.i  abed  means  cd  multiplied 
by  b,  and  that  product  by  a. 

57.  Law  of  signs.  From  the  definition  it  appears  that 
like  signs  produce  plus,  and  unlike  signs  minus. 

I.e.,  +  X  +  =  + 

+  X  -  =  - 

-  X  4-  =  - 

-  X  -  =  + 

58.  Reading  of  products.  As  already  stated,  the  original 
meaning  of  the  word  times  referred  to  positive  integers. 
The  expressions  f  times,  ^  of  a  time,  and  —  2  times  are 
meaningless  in  the  original  sense  of  the  word.  But  with 
the  extension  of  the  definition  of  multiplication  has  come 
an  extension  of  the  meaning  of  the  word  times,  so  that  it 
is  now  generally  used  for  all  products,  as  in  §  53. 

Thus,  the  expression  2^  times  as  much  is  generally  used,  although  it 
is  impossible  to  pick  up  a  book  2^  times.  So  (—  2)  x  (—  3)  is  read, 
"  minus  2  times  minus  3,"  although  we  cannot  look  out  of  a  window 
—  2  times. 

As  already  stated,  the  word  into  is  sometimes  used  in 
algebra  to  indicate  the  product  of  two  or  more  factors, 
(—  a)  (—  b)  being  read  "  —  a  into  —  Z»." 

The  parentheses  about  negative  factors  are  omitted  when 
no  misunderstanding  is  probable.  Thus,  (—  a)  •  (—  b)  may 
be  written  —  a  X  —  b,  or  even  —a  —  b.  But  —  a^  and 
(—  ay  are  not  the  same,  the  former  meaning  —  aa  and  the 
latter  —  a-  —  a,  oi  4-  a^. 


MULTIPLICATION. 


43 


59.  The  Associative  and  Commutative  Laws  of  Multiplica- 
tion. Before  we  are  able  to  proceed  with  certainty  in  mul- 
tiplication, it  is  necessary  to  show  that  we  can  change  the 
order  and  grouping  of  the  factors  to  suit  our  convenience. 

For  example,  to  prove  that  a6c,  which  by  definition  means  a(6c), 
=  {ab)c  =  {ac)b  =  b{ac)  •  •  • . 

Proof.  1.  Suppose  this  solid  to  be  composed  of  inch 
cubes,  and  to  have  the  dimensions  4  in.,  5  in.,  6  in. 

2.  Then,  since  there  are  4  cubes  in 
the  row  OA,  and  there  are  5  such 
rows  in  the  layer  CA,  there  are  (5  •  4) 
cubes  in  that  layer.  And  since  there 
are  6  such  layers,  there  are  6  •  (5  •  4) 
cubes  in  all. 

3.  Similarly,  since  there  are  6  in 
column  OB,  and  there  are  4  such 
columns  in  layer  BA,  there  are  (4  •  6) 
cubes  in  that  layer.     And  since  there 

are  5  such  layers,  there  are  5  •  (4  •  6)  cubes  in  all. 

4.  Similarly,  there  are  4  •  (5  •  6)  cubes. 

5.  But  the  total  number  is  the  same, 

.-.6 -(5 -4)=  5 -(4. 6)  =  4.  (5. 6). 

6.  And  since  the  proof  is  independent  of  the  numbers, 
.' .  a  ■  (b  •  c)  =  b  •  (g  '  a)  ^  c  •  (b  •  a)  =  (a  •  b)  •  c  =  •  •  • . 

7.  By  taking  d  such  solids  it  could  be  proved  that 

a-  (b  •c-d)  =  (a-b)  ■  (c-d)  =  (a'b-cy'd  =  b-  a-  (d-c)=  •  •  • , 
and  similarly  for  any  number  of  letters. 

8.  And  since  in  multiplications  involving  negative  num- 
bers we  proceed  as  if  the  numbers  were  positive,  prefixing 
the  proper  sign,  therefore  the  proof  is  general  for  all 
integers. 


44  ELEMENTS   OF  ALGEBRA. 

EXERCISES.    XVII. 
Perform  the  multiplications  indicated  : 


1. 

-2.-7. 

2. 

-4.-3. 

3. 

72.-^.-3. 

4. 

-h'-h--h 

5. 

(-2)^.  (-3)1 

6. 

(-iy^.(-2y. 

7. 

4.5.-3.2.1.1. 

8. 

l._2.3.-4.5.-6. 

9. 

-1.-2. -3. -4. 

10. 

5.3.1._1._3._5. 

1. 

l.(_  2)2.33.  (-4)^ 

12. 

(_  1)100.  (_l)99.^_  2)5. 

3. 

4. 3. 2. 1.0.-1.-2. 

-3 

-4. 

60.  The  index  law.  Since  a^  =  aa,  and  a^  =  aaa,  there- 
fore a^  ■  a^  =  aa  •  aaa  =  a^.  Similarly,  if  m  and  n  are  posi- 
tive integers, 

a™  =  aaa  ■  ■  •    to  m  factors, 

and  a"  =  aaa ...    to  n       " 

a"* .  a-**  =  aaaa  •  ■  •  to  m  -^  n       " 

This  is  known  as  the  index  law  of  multiplication. 

Hence,  2  a^b^^c^  ■  5  a^ftV  =  10  a^b^c^. 

The  cases  in  which  m  and  n  are  negative,  zero,  or  fractional  are 
considered  later. 

EXERCISES.    XVIII. 

Perform  the  multiplications  indicated  : 

1.  -a^.(^-ay. 

2.  25ab^G^d^'2a^bh''d. 

3.  —  a  '  —  a^  ■  —  a^  ■  —  a^ '  —  a^. 

4.  -a-(-ay-(-ay-(~ay-(-ay. 

5.  x'^  -x^  .  X. 

6.  £c"*?/"  •  ic"?/'"  .  x^y"^. 

7.  x^y^z^  .  £cy2^3  •  xHj^z^  •  xyz. 

8.  a:^'^ .  a:;^ .  x^  -  y^  ■  y^  •  y'^  •  z^  -  z^  •  z. 


MULTIPLICATION.  45 

II.     MULTIPLICATION  OF  A  POLYNOMIAL  BY  A  MONOMIAL. 

61.    I.    When  the  monomial  is  a  positive  integer,  as  in  the 
case  of.  a(b  —  c). 

1.  '.'  a  =  l-{-l-\-l-\----toa  terms, 

2.  .'.  a(b  —  c)  =  (b  —  c)H-(6  —  c)-\-(b  —  c)-\ to  a  terms, 

Def.  mult.  §  54 

3.  =  b  -^  b  -\-  b  -{-  •■•  to  a  terms, 

—  c  —  G  —  c  —  "-toa  terms,   §  49 

4.  =  ab  —  ac.  Def.  mult.  §  54 
E.g.,  2{x-i-y)  =  {x  +  y)  +  {x  +  y)  =  2x  +  2y. 


II.    When  the  monomial  is  a  positive  fraction,  as  in  the 

0- 

X      l  +  lH-l  +  ---toa!  terms 


case  of  -  (b  —  c). 


1.  •.• 


y 


,   ic  _  (b  —  c)  -\-(b  --  g)-\-  ■  ■■  to  X  terms 

Zi.   •  .      (o        c )  ^^  ■  > 

^  -^  Def.  mult.  §  54 

xb  —  XG  .     _ 

6.  = }  as  m  I 

y 

4.  = }  because  xb  ?/ths  minus  xc  yths 

y       y  ./  J 

is  the  same  as  (xb  —  xc)  ^/ths. 

III.    When  the  monomial  is  negative,  and  either  integral 
or  fractional,  as  in  the  case  of  (—  m)  (b  —  c). 

1.  '.'  —  m  =  m  •  1,  preceded  by  the  sign  — , 

2.  .'.  (—  m)(b  —  c)  =  m  (b  —  c)  preceded  by  the  sign  — , 

Def.  mult.  §  53 

3.  =  (mb  —  mc)  preceded  by  the  sign  — , 

I  and  II 

4.  =  —  m^  +  mc.  §  46 


46  ELEMENTS   OF  ALGEBRA. 

62.  From  the  results  of  these  three  cases  it  appears  that : 
To  multiply  a  polynomial  by  a  monomial  is  to  multiply 

each  term,  of  the  polynomial  by  the  monomial  and  to  add 
the  products. 

Since  the  multiplier  is  distributed  among  the  terms  of 
the  multiplicand,  this  statement  is  known  as  the  distribu- 
tive law  of  multiplication. 

E.g.,  3  a2  (a*  -  6)  =  3  ««  -  3  oPh.  This  can  be  checked  by  letting 
a  =  1,  6  =  2.  Then  3  a2  («!  _  &)  =  3  .  _  1  ^  _  3^  and  3  a^  -  3  a26  = 
3  _  6  =  -  8. 

EXERCISES.    XIX. 

Perform  the  following  multiplications,  checking  the  re- 
sults by  assigning  arbitrary  values  to  the  letters : 

1.  a''{a^-\-b^-G'-). 

2.  5  m^xy  {xz^  —  S  z^x  —  4). 

3.  -7x''y(-Sxy^  +  2xy). 

4.  25ab^c^d\2a'b^-i-2c''d). 

5.  -5a[-3a  +  2(a-2)]. 

6.-7  m^n^  (2  m  —  3  ?i  —  4  mn  +  6  m^n). 

III.     MULTIPLICATION   OF  A  POLYNOMIAL  BY  A 
POLYNOMIAL. 

63.  Eequired  the  product  of  (a  -j-  6)  (c  +  ^). 

1.  Let  m  =  (a  +  b). 

2.  Then  m{G  +  d)  =  mG  -\-  md,  §  61 

3.  =  (a  +  b)c  +{a  -\-  b)d,  •  .•  m  =  {a  +  b) 

4.  =  ac  +  be -\- ad  +  bd.  §§51,61 

From  this  it  appears  that  to  multiply  one  polynomial  by 
another  is  to  multiply  each  term  of  the  first  by  each  term  of 
the  second  and  to  add  the  products. 

This  is  the  general  form  of  the  distributive  law  of  multiplication. 


MULTIPLICATION.  47 

The  following  example  illustrates  the  process : 

x^  -\-2xy   +y^ 

^  +  y 

Product  by  x,        x^  +  2  x^y  +     xy^ 
Product  by  y,  x^y  +  2  xy^  +  y^ 

Sum  of  products,  cc^  +  3  x^y  +  3  xy^  +  y^ 

.'.  (X  +  7j)  (x'  +  2xy  +  7/2)  =  x'  +  3x'y  +  3xy'-\-  y\ 
Check.     Let  x  =  1,  y  =  1.     Then 
1+2+1=4 
1  +  1  =  2 
1  +  3  +  3  +  1  =  8,  or  2. 4. 

Since  the  identity  liolds  true  for  any  values  of  x  and  ?/,  it  holds  true 
\l  X  =  y  =  \^  as  in  the  above  check.  It  is  evident,  however,  that  the 
value  1  does  not  check  tlie  exponents.  Where  there  is  any  doubt  as 
to  these,  other  values  must  be  substituted. 

EXERCISES.    XX. 

Perform  the  following  multiplications,  checking  the  re- 
sults by  assigning  arbitrary  values  to  the  letters. 

1.   {a  +  h){x  +  y).  2.   (x  +  y){x-y). 

3.   {x^  -  y^)  {x  +  y).  4.    {p^  +  q'')  {x^  -  3  /). 

7.  (a3  +  a2  +  a  +  l)(a-l). 

8.  {bx^-x  +  l){3x^-x-2). 

9.  {2x  +  3y -z){2x-3y +  z). 

10.  (ic*  +  £c3  +  x^  +  a;  +  1)  (x  —  1). 

11.  (x  +  y){x^  +  3xhj  +  3xy''  +  y^). 

12.  (3  ^2  _  2  a)  (5  a«  -  2  ^2  _  3  ^  _p  4)_ 

13.   {a-  b)  (a'  +  a^b  +  a^b""  +  a^b^  +  a^b''  +  a^'b^  +  ab^  +  ^^). 


48  ELEMENTS   OF  ALGEBRA. 

64.  A  polynomial  is  said  to  be  arranged  according  to  the 
powers  of  some  letter  when  the  exponents  of  that  letter  in 
the  successive  terms  either  increase  or  decrease  continually. 

In  the  former  case  the  polynomial  is  said  to  be  arranged 
according  to  ascending  powers,  in  the  latter  according  to 
descending  poivers  of  the  letter. 

E.g.^  x^  +  3x3  +  a;2  4-  1  is  arranged  according  to  descending  powers 
of  X.  If  it  is  desired  to  have  all  of  the  powers  represented,  it  is  written 
x5  +  0x4  +  3x3+x2  +  0x  +  1. 

The  polynomial  x^  —  3  x^y  +  3  xy^  —  y^  is  arranged  according  to 
descending  powers  of  x  and  ascending  powers  of  y. 

There  is  evidently  an  advantage  in  arranging  both  multi- 
plicand and  multiplier  according  to  the  powers  of  some 
letter,  as  shown  by  the  following  example : 

Not  Abkanged.  Arranged, 

2/2  +  x2  +  2  x?/  x2  +  2  x?/  +  2/^ 

x  +  y  x-\-y 


xy-i  +  x3  4-  2  x2?/  x3  +  2  x^y  +    xy^ 

+y3  +  x2y  +  2x?/2  x2y  +  2  X?/2  -t-  y3 

X2/2  +  X^  +  2  X^Z/  +  y3  ^  X2y  +  2  X?/2  X3  +  3  X2?/  +  3  X?/2  -f  2/3 

=  x3  +  3x2?/  +  3a;y2  _!_  yz  Check.     Let  x  =  1,  ?/  =  1.     Then 

2-4  =  8. 

The  method  at  the  right  is  evidently  much  simpler. 

65.  It  is  also  evident  that  the  product  of  the  terms  of 
highest  degree  in  any  letter  in  the  factors  is  the  term  of 
highest  degree  in  that  letter  in  the  product.  Also  that  the 
product  of  the  terms  of  lowest  degree  in  any  letter  in  the 
factors  is  the  term  of  lowest  degree  in  that  letter  in  the 
product. 

Hence,  if  the  factors  are  both  arranged  according  to  the  descending 
(or  ascending)  powers  of  some  letter,  the  first  term  of  the  product  will 
be  the  product  of  the  first  terms,  and  the  last  term  will  be  the  product 
of  the  last  terms. 


MULTIPLICATION.  49 


EXERCISES.    XXI. 


Perform  the  following  multiplications,  checking  the  re-' 
suits  by  assigning  arbitrary  values  to  the  letters : 

1.  x^  —  if  by  x^  +  if. 

2.  a?-x  +  x^a  by  x^a  —  a^x. 

3.  x^y'^  —  x^  —  y^  by  y  —  x. 

4.  X  -\-  y  -\-  z  hj  x  -\-  y  —  z. 

5.  1  -  a2  +  a*  -  a^  by  1  +  a'^- 

6.  x'^  +  xhj  -\-  y^  by  cc^  —  3  a?  +  ?/. 

7.  i  2/'  -  ^  2/^  +  i  ^'  by  ^  7/  -  ^  ;?;. 

8.  xyz  —  x^  —  y"^  —  z^  by  a?  +  ?/  +  ^. 

9.  p'^  —  2pq  +  ^^  by  ^^  +  2^2'  +  q"^.  • 

10.  -  ^2  +  3  a^*  +  52  by  ^ah-h''  +  a\ 

11.  «,^  —  a*  +  a^  —  a^  -^  (X  —  1  by  a  +  1. 

12.  ic^  —  3  ic^?/  -\-  3  o:;?/^  —  2/^  by  x^  —  2xy  -\-  y^. 

13.  cc?/  +  2  cc^  —  3  2/^  +  ic^  +  ?/^  +  4  ^^  by  x  —  y  —  2  z. 

66.  Detached  coefficients  may  be  employed  in  multiplica- 
tion whenever  it  is  apparent  what  the  literal  part  of  the 
product  will  be. 

E.g.^  in  multiplying  x^  -\-  pz  ■\-  q  hy  x^  —  x  -\-  pq  the  coefficients 
cannot  be  detached  to  advantage. 

But  in  multiplying  x^  +  2  xy  +  2/^  hy  x  +  8  y,  it  is  apparent  that  the 
exponents  of  x  decrease  by  1  while  those  of  y  increase  by  1  in  each 
factor,  and  that  this  law  will  also  hold  in  the  product.  Hence,  when 
the  coefficients  are  known  the  product  is  known  also,  and  the  multi- 
plication may  be  performed  as  follows  : 

Check. 
1+2+1  =4 

1  +3  =4 

1  +  2  +  1 

3  +  6  +  3 
1  +  5  +  7  +  3  =.16 

.-.  (X  +  Zy)  (x2  +  2x2/  +  2/^)  =  x3  +  5x2y  +  7x?/2  +  3y3. 


50  ELEMENTS   OE   ALGEBRA. 

67.  If  the  coefficient  of  the  first  term  of  the  multiplier  is 
1,  as  is  frequently  the  case,  the  work  can  be  materially 
simplified  by  the  following  arrangement: 

The  problem  is  the  same  as  the  preceding  one. 


1 
+  3 


1+2  +  1  Check.     4  •  4  =  16. 

3  +  6  +  3 


1  +  5  +  7  +  3  x3  +  5x22/  +  7ay2  +  3i 


68.  In  case  any  powers  are  lacking  in  the  arrangement  of 
the  polynomial,  zeros  should  be  inserted  to  represent  the 
coefficients  of  the  missing  terms. 

E.g..,  to  multiply  x"^  -{■  xy  -\-  ip-  by  x^  +  i/^,  either  of  the  following 
arrangements  may  be  used : 

1  +  1  +  1     Check.     2-3  =  6. 


1+1+1  1 

1+0+1  +0 

1+1+1  +1 


1  +  1  +  1 


1+1+2+1+1 
1  +  1  +  1 


1  +  1+2  +  1  +  1  X*  +  x^?/  +  2  x2?/2  +  x?/3  +  2^ 

EXERCISES.    XXII. 

Perform  the  multiplications  indicated  in  exs.  1-13,  by 
detached  coefficients,  checking  the  results  as  usual. 
P  =  cc^  —  x^y  ■\-  xy^  —  2/^     Q  ^  ^  —  V)    R  ^  x'^  —  xy  -{-  y^. 
1.    PQ,  2.    PR.  3.    QR.  4.    P\ 

5.    Q-'R.  6.    R\  7.    QR\  8.    Q^R\ 

9.    {x  +  yf.  10.    {x-yy.  11.    {x -{- yf. 

12.    (x-ijy.  13.    (x  +  y)(x-y). 

14.  Verify  the  following  identities,  (1)  by  substituting 
arbitrary  values,  (2)  by  expanding  both  sides  of  the  iden- 
tity, using  detached  coefficients  or  not  as  seems  best : 

(a)  (x  +  y  +  zy-  (x^  +  2/'  +  ^')  =  3(a5+-7/)  (y  +  z)  (z  +  x). 

(b)  (x-\-yy-\-(y  +  zy+(z  +  xy-(x'  +  y'  +  z')-=(x+y+zy. 


MULTIPLICATION. 


51 


IV.  SPECIAL  PRODUCTS  FREQUENTLY  MET. 

69.  In  exs.  9-13  on  p.  50  five  products  were  found  which 
are  so  frequently  used  as  to  require  memorizing.  They  are 
as  follows : 

1.   (cc  +  2/)^  =  ^^  +  2  ic?/  4-  ?/^      Hence,  the  square  of  the 
sum  of  two  numbers  equals  the  sum  of 
their  squares  plus  twice  their  product. 

This  theorem  may  be  illustrated  by  a  figure. 
Here  the  square  ^.0  =  (x  +  ?/)2,  the  square 
AP  =  x^,  the  square  PC  =  y^,  and  there  are  two 
rectangles  equal  to  EF  =  xy.     And 

:•  AC  =  AP  +  2EF+PC, 
.-.  (a;  +  2/)2  =  x2+2xy  +  ?/2. 


n 

G 

H 

y 

xy 

y 

r 

X 

p  y 

X 

x" 

X 

X 

xy 

y 

B 


xy 


^< 


2.   (x  —  yy  =  x'^  —  2xy-\-  y^.     Hence,  the  square  of  the 
difference  of  two  numbers  equals   the 
sum  of  their  squares  minus  twice  their 
product. 

In  the  figure,  AP^  =  x%  BH  =  y^,  AC  = 
{X  -  y)2,  and  DP=  CH  =  xy.     And 

:•  AC  =  AP-2DP-\-BH, 

.-.  {X  -y)^^x^  -2xy  +  y^. 
Expressions  of  the  form  x  -\-  y,  x  —  y,  are 
called  conjugates  of  each  other. 


(x-y)' 


y. 


H 


3.   (x  +  y)  {x  —  y)  =  x^  —  y\     Hence,  the  product  of  the 
two  conjugate  binomials  equals  the  differ- 
ence of  their  squares  (i.e.,  the  square   of 
the  minuend  minus  the  square  of  the  sub- 
trahend). 

In  the  figure,  AG  =  x^,  AF  =  y^,  and  GG  -\-  FB 
=  x{x-y)  +y{x-y)  =  {x  +  y){x-  y).     And 
•.•  GG  +  FB  =  AC  -AF, 

.-.  (.T  +  2/)  (X  -  ?/)  =  X2  -  2/2. 


x-y  ^^"--y^ 

F 

y     y^  y 
y 

52  ELEMENTS   OF  ALGEBRA. 

4.  (x  -\-  yy  =  x^  -\- 3  x^y  +  3  xy'^  +  y^.  Hence,  the  cube  of 
the  sum  of  two  numbers  equals  the  sum  of  their  cubes  plus 
three  times  the  square  of  the  first  into  the  second  plus  three 
times  the  square  of  the  second  into  the  first. 

5.  (x  —  yy  =  x^  —  3  x^y  +  dxy"^  —  y^.    (State  the  theorem.) 

EXERCISES.    XXIII. 

By  the  help  of  the  theorems  of  §  69  expand  the  expres- 
sions in  exs.  1-18. 

1.  42  X  38,  i.e.,  (40  +  2)  (40  -  2). 

2.  23  X  17.  3.  95  X  85. 
4.  (^2  +  3)2.  5.  (a^-2)2. 
6.  (2^  +  1)'.                                       7.  (2cc2-l)3. 

8.  (2  x""  -  yf.  ■  9.  [a-(b  +  c)^. 

10.  (2cc2  +  l)(2cc2_l).  11.  (a2  +  3)(a2_3). 

12.  (a-]-b-ab)(a  +  b-^  ab).         13.  [(a  +  b)  {a  -  b)J. 

14.  {a''-\-2ab-{-b'){a''-2ab  +  b'').   15.  (x'' +  y')  (x^  -  y^) . 

16.  42^.  17.    49-51.  18.    492. 

19.  Verify  the  following  identities  : 

(a)  (^2  +  ^,2  _^  c2  +  ^2)  (^2  _|_  ^2  _^  ^2  _^  ^2^  _  ^^^  +  &ic  + 

c?/  +  cZ«)^  =  {ax  —  bwy  +  {cz  —  dyy  +  («?/  —  cw)^  +  {dx  — 
bzy  +  {az  -  dwy  +  (6?/  -  cxy. 

(b)  (a?  +  2/)^  —  £c^  —  ?/^  =  3  xy  {x  +  ?/)  (a?^  -\- xy  -\-  yy. 

(c)  (x  +  ij)^  -x^  -i/  =  5  xyix  +  ^)  {f  +  xy  +  y^. 

(d)  (a;  +  yy  -x'  -y'  =  lxy{x  +  y)  (rr^  +  a:^/  +  y^. 

20.  Expand  the  following  expressions  by  the  help  of  the 
theorems  of  §  69,  checking  by  arbitrary  values : 

(a)  {x^  +  yy.  (b)  {x'  +  yy. 


MULTIPLICATION.  63 

V.     INVOLUTION. 

70.  The  product  of  several  equal  factors  is  called  a  power 
of  one  of  them  (§8). 

The  broader  meaniug  of  the  word  power  is  discussed  later  (§  130). 
At  present  the  term  will  be  restricted  to  positive  integral  power. 

71.  The  operation  of  finding  a  power  of  a  number  or  of 
an  algebraic  expression  is  called  involution. 

The  student  has  already  proved  one  important  proposition  in  in- 
volution, viz.,  that  a"*  •  a'*  =  a"*  +  »*,  where  the  exponents  are  positive 
integers  (§  60). 

He  has  also  learned  how  to  raise  the  binomial  x  ±y  to  the  second 
and  third  powers  {§  69). 

It  now  becomes  necessary  to  consider  certain  other  theorems. 

72.  Notation.     If  m  and  n  are  positive  integers, 
(aJ^y  means  a^  •  a"^  •  a™'  ■  •  •  to  n    factors,  each  a^ ; 

«"•"        "       a-a-a--  •        to  m"       "  "      a. 

E.g.,  (a3)2  means  a^  •  a^  =  a^  +  s  =  aS  . 

of'       "      a  •  a  •  a  •  •  •  to  32  factors,  =  a^ ; 
of       "      a- a- a- ■■  to  2^      "       =aP. 

73.  d^  has  already  been  defined  to  equal  a. 

74.  The  expression  a^,  a  being  either  positive  or  negative, 
is  defined  to  equal  1,  for  reasons  hereafter  set  forth  (§  214). 

75.  Theorem.  The  nth  power  of  the  m.th  power  of  an 
algebraic  expression  equals  the  math  power  of  the  expression. 

Given        an  algebraic  expression  a,  and  m  and  n  positive 

integers. 
To  prove   that  (a/^y  =  a""^. 
Proof.  1.   (a™)"  means  a""  •  a""  ■  a""  •  •  ■  to  n  factors,  each  a"*, 

2_  =   Q^m  +  m  +  m+  .  •  .to7i  terms,  each  m  ff   QQ 

3.  =  a'"". 


64  ELEMENTS   OF  ALGEBRA. 

76.  Theorem.  The  mth  power  of  the  product  of  several 
algebraic  expressions  equals  the  product  of  the  jnth  powers 
of  the  expressions. 

Given        the  expressions  a,  b,  c,  -  •-,  and  m  an  integer. 
To  prove   that  (abc  •••)"•  =  a"'b"'c"'  •  •  • . 

Proof.  1.  (abc  •  •  •)'"  means   (abc  •  •  •)  •  (abc  •  •  •)  •  (abc  •••)•••, 
to  m  groups,  each  (abc  -  ■  •) 

2.  =  (aaa  •  •  •  to  m  factors)  •  (bbb  •  •  •  to  m  factors)  •  (ccc 
■  ••  to  m  factors)  •  •  •  §  59 

3.  =  a'^b'^c"*.  -  Def.  of  power 

77.  Law  of  signs.     Since 

-j-  a-  -\-  a  =  -\-  a^, 
and  —  a  ■  —  a  =  -\-  a"^, 

but  —  a-  —  a-  —  a  ^i  —  a^, 

it  is  easily  seen  that 

1.  Powers  of  positive  expressions  are  positive  ; 

2.  Even  powers  of  negative  expressions  are  positive  ; 

3.  Odd  powers  of  negative  expressions  are  negative. 

EXERCISES.    XX  JV. 

Express  without  parentheses  exs.  1-12. 
1.   (a'^x'^y.  2.   (aH"^)"".  3.   (a%^c^d^y. 

4.   (-a'^b^'cy.  5.    (-ab'^cy.  6.   -(a^^^V)*. 

7.   (ay,  (ay.  8.   (a^,  (a^.  9.   (-  a'«^.«)2'»«. 

10.   (-^a^'^y^.       11.   -[-(ayy.         12.   -(-a^^b^cy. 

13.  Prove  that  (a'^y  ^  (a'^y. 

14.  Is  it  true  that  a"""  =  a"'"  ?     Proof. 

15.  Also  that  (a'^b'^y  =  (a"Z>'»)'««  ?     Proof. 


MULTIPLICATION.  55 

78.  Powers  of  polynomials.  A  polynomial  can  be  raised 
to  any  power  by  ordinary  multiplication. 

But  in  raising  to  the  4tli  power  it  is  easier  to  square  and 
then  to  square  again,  since  (cl'^Y  =  a^. 

E.g.,  to  expand  {x  —  2yy. 

1.  (x-2y)2  =  x2-4x2/  +  4y2.  §69 

2.  (x2  -  4  X2/  +  4  2/2)2  =  [-(x2  _  4  xy)  +  4  ?y2]2  §  46 

3.  =  (x2  _  4  x?/)2  +  2  (x2  -  4  x?/)  •  4  2/2  +  16  y^ 

4.  =x4  -  Sx^y  +  16x2?/2  +  8x2y2  _  S2xy^  +  IB?/* 

5.  =x''-8x32/  +  24x2?/2-32x?/3  +  16y4. 
Check.  (-  1)4  =  1  -  8  +  24  -  32  +  16  =  1. 

Similarly,  to  raise  to  the  6th  power  first  cube  and  then 
square,  since  (a^y  =  a^. 

But  to  raise  to  the  5th,  7th,  11th,  or  other  powers  of 
prime  degree,  multiply  out  by  detached  coefficients. 

EXERCISES.    XXV. 

Expand  the  expressions  in  exs.  1-20. 

1.  (20  +  1)2.  2.  (x'--3  7/y. 

3.  (x  +  3yy.  4.  (2x-7yy. 

5.  (x""  +  2/")*.  6.  (a  +  b  +  cy. 

7.  (^x-^yy.  8.  (-x-3yy. 

9.  (2x^-3yy.  10.  (a -h  2  b -hey. 

11.  (-a-b-cy.  12.  (a^-h2ab  +  by. 

15.    (a^o-b'^ay.  16.    (x^  +  xY  +  y^. 

17.    (ia  +  2b  +  cy.  18.    (31  m^  -  20 7^2)2. 

19.    (a  -  2  ^  4-  3  c)2.  20.    (a  —  b  +  c-  dy. 


56  ELEMENTS   OF   ALGEBRA. 

79.  The  Binomial  Theorem.  It  frequently  becomes  neces- 
sary to  raise  binomials  to  various  powers.  There  is  a  simple 
law  for  effecting  this,  known  as  the  Binomial  Theorem. 

The  student  will  discover  most  of  this  law  in  answering  the  following 
questions : 

Expand  (a  +  by,  (a  +  hf,  {a  +  by,  {a  +  by. 

(a)  How  does  the  number  of  terms  in  each  expansion 
compare  with  the  degree  of  the  binomial  ? 

(b)  How  do  the  exponents  of  a  change  in  the  successive 
terms  ? 

(c)  How  do  the  exponents  of  b  change  in  the  successive 
terms  ? 

(d)  In  each  case,  what  is  the  first  coefficient  ?  How  does 
the  second  coefficient  compare  with  the  exponent  of  the 
binomial  ? 

(e)  In  the  case  of  the  4th  power  does  the  third  coefficient 
equal  — r-  ?  In  the  5th  power  is  it  —^  ?  What  will  it 
probably  be  in  the  6th  power  ?    in  the  7th  ?    in  the  7ith  ? 

(f )  In  the  case  of  the  4th  power  does  the  fourth  coeffi- 

4  •  3  •  2  5  •  4  •  3 

cient  equal  ?    In  the  5th  power  is  it         ^     ?     What 

will  it  probably  be  in  the  6th  power  ?    in  the  7th  ?    in  the 
nth? 

(g)  In  the  case  of  the  4th  power  does  the  fifth  coefficient 
equal         „    '     ?    In  the  5th  power  is  it  ?    What 

will  it  probably  be  in  the  6th  power  ?    in  the  7th  ?    in  the 
nth.  ? 

( h)  In  expanding  (a  +  by,  what  will  be  the  coefficient  of 
a%  ?  of  a^b^  ?  (The  student  should  now  be  able  to  answer 
without  actual  multiplication.) 


MULTIPLICATION.      .  57 

80.  Theorem.  If  the  binomial  a  -\-  b  is  raised  to  the  nth. 
power,  71,  integral  and  positive,  the  result  is  expressed  by 
the  formula 

(a  +  by  =  a"  +  na^'-'^b  +  ^^^^^~    ^  a"-^b^ 


where  : 


2 
23 


^.(. -!)(.- 2)  ^^„_3^3^ 


1.  The  number  of  terms  in  the  second  member  ^s  n  +  1 ; 

2.  The  exponents  of  a  decrease  from  n  to  0,  while  those  of 
b  increase  from  0  to  n; 

3.  The  first  coefficient  is  1,  the  second  is  n,  and  any  other 
is  formed  by  multiplying  the  coefficient  of  the  preceding 
term  by  the  exponent  of  a  in  that  term  and  dividing  by  1 
more  than  the  exponent  of  b. 

The  proof  of  this  theorem,  which  has  already  been  found  inductively 
on  p.  56,  may  be  taken  now  or  it  may  be  postponed  until  later  in  the 
course.     The  proof  is  given  in  Appendix  I. 

81.  Pascal's  Triangle.  The  coefficients  of  the  various 
powers  of  the  binomial  f+n  are  easily  found  by  a  simple 
arrangement  known  as  PascaVs  Triangle,  from  the  famous 
mathematician  who  made  some  study  of  its  properties. 

Coefficients  for  1st  power    1     1 


"  2d 

2  1 

"  3d 

3  3  1 

" .  4th 

4  6  4  1 

"  5th 

5  10  10  6 

"  6th 

6  15  20  15 

Each  number  is  easily  seen  to  be  the  sum  of  the  number  above  and 
the  number  to  the  left  of  the  latter. 

Write  down  the  coefficients  for  the  7th,  8th,  9th,  and  10th  powers, 
thus  enlarging  Pascal's  triangle. 

For  note  on  Pascal,  see  the  Table  of  Biography. 


58  ElyEMENTS   OF  ALGEBRA, 

82.  Various  powers  of  f  +  n.  These  are  needed  in  the 
extraction  of  roots  (§§  128-133)  and  should  be  verified  by 
the  student. 

(/  +  nf  ^fJ^^fH  +  Zfu"  +  n\ 

(/  +  ny  =r  +  4.fn  +  e/^Ti^  +  4>^  +  n\ 

(/+  ny  =  (Expand  it.) 

(f-^ny=  "        " 

{f+ny=  "         « 

Illustrative  problems.     1.  Expand  (2  a  —  3  b'^y. 

1.  (2a -  362)3 ^  (2a)3  +  3 (2a)2(-  362)  +  3 (2a)  (-  352)2  +  (_  352)3 

2.  =  8  a3  -  36  a262  +  54  a64  -  27  6^. 

Check.     (-l)3.=  8-36  +  64-27=: -1. 

In  cases  like  this  it  is  better  to  indicate  the  work  in  the  first  step 
and  then  simplify. 

2.  Expand  (|  -  2/  +  ^')'' 

2.  =(|-2/)%2(|-2/);22+(2;2)2 

3.  ■  = xy  +  ?/2  +  X2;2  -  2  2/z2  +  z*. 

4 

EXEBCISES.    XXVI. 

Expand  the  following  expressions  : 
1.   (x  +  yy.  2.    (1-ay. 

3.   (x^  —  yy.  4.   (x  —  yy\ 

5.   (a-2Z')2.  6.   (2CC  +  2/')'- 

7.   {x^y-Zy^y.  8.   (a;  +  3/ -  «)2. 


MULTIPLICATION.  59 

11.   {a-h-cy.  12.   (2a -^ly. 

13.   {a-b  +  cy.  ^  14.   {^x  +  2yy. 

15.   {^^''-iy'^Y'  16.   (3a2_2a6  +  62)8. 

"■(-0'  ■      ■•(-9' 


REVIEW   EXERCISES.    XXVII. 

1.  Solve  the  equation  184  —  cc^  =  40.     Check. 

2.  What  is  the  etymological  meaning  of  multiply?  of 
abstract  ?   of  ascending  ?   of  descending  ?    of  coTnmutative  ? 

3.  Show  that  the  arithmetic  definition  of  multiplication 
is  not  broad  enough  for  algebra.  Explain  the  de^nition 
in  §  54. 

4.  What  is  the  broader  meaning  of  the  word  times  in 
algebra  ?     Illustrate. 

5.  What  is  the  Index  Law  of  multiplication  ?  Has  it 
been  proved  by  you  for  all  kinds  of  exponents  ?  If  not, 
for  what  kind  ?     Prove  it. 

6.  What  is  meant  by  the  Distributive  Law  of  multipli- 
cation ?     Prove  the  law. 

7.  Make  up  an  example  illustrating  the  advantage  of 
arranging  the  terms  according  to  the  powers  of  some  letter 
in  multiplication. 

8.  What  are  the  advantages  in  using  detached  coefficients 
in  multiplication  ?     Illustrate  by  solving  a  problem. 


CHAPTER   IV. 

DIVISION. 

I.     DEFINITIONS   AND  LAWS. 

83.  Division  is  the  operation  by  which,  having  the  product 
of  two  expressions  and  one  of  them  (not  zero)  given,  the 
other  is  found. 

Thus,  6  is  the  product  of  2  and  3  ;  given  6  and  2,  3  can  be  found. 

The  given  product  is  called  the  dividend,  the  given  expres- 
sion is  called  the  divisor,  and  the  required  expression  is 
called  the  quotient. 

84.  Since  0  =  a  •  0  (§  55),  it  follows  that  -  should  be 
defined  to  mean  0. 

85.  Law  of  signs.     Since 

-\-  a  ■  -\-  b  ^  +  ab, 

-\-  a  •  —  b  ^  —  ab, 

—  a  ■  -}-  b  =  —  ab, 
and  —  a-  —  b  =  -\-  ab, 

it  therefore  follows,  from  the  definition  of  division,  that 
■i-  ab  -T-  +  a  =  -{-  b, 

—  ab  -i-  -{-  a  =  —  b, 

—  ab  -. a  =  +  5, 

and  -\-  ab  -. a  =  —  b. 

That  is,  like  signs  in  dividend  and  divisor  produce  +, 
and  unlike  signs  — ,  in  the  quotient. 


DIVISION.  •  61 

86.  Index  law.  Since  o^"*~''  •  a"  =  o^"»,  by  the  index  law  of 
multiplication  (§  60),  therefore,  —  =  a™""",  by  the  definition 
of  division. 

Hence,  10  a%H^  -f-  5  oFbH'^  =  2  a^¥c^. 

The  above  proof  is  based  on  the  supposition  that  m>n,  and  that 
both  are  positive  integers.  The  cases  in  which  m  and  n  are  zero, 
negative,  and  fractional,  and  in  which  m<n,  are  considered  later. 


EXERCISES.  XXVIII. 

Perform  the  following  divisions  : 

1.    -  125  --  -  25.  2.    80  --  -  16. 

3  a%e  25  a'bc' 

3. —  4.    3-T7-- 

—  oc  —  o  a^OG 

-  10  xy^z^  49  a;^yQ^V 
^*      -5  2/V   '  ^'       -Iwz^ 

-56a^'b^'c  -27  (a -b) 

—  8  a^%G  '          a  —  b 


II.     DIVISION   OF  A  POLYNOMIAL  BY  A  MONOMIAL. 

87.    1.  •.'  ma  -\-  mb  -^  mo  =  m  (a  -\-  b  +  c).  §  61 

„         ma  -{-  mb  -\- mc  ,    ,    ,  -r.  .c     ^  j  •    ■  • 

2.  .*. =  a  -\-  b  +  s.       Del  oi  division 

m 

3.    Hence,  to  divide  a  polynomyial  by  a  monomial  is  to 

divide  each  term,  of  the  polynoinial  by  the  monoinial  and  to 

add  the  quotients. 

Thus,  ^^!-^ ^^^^  =  2a-h.     Check.    Let  a  =  2,  6  =  3.     Then 

-  17  a62 

17  ■  2  .  27  -  34  •  4  ■  9      „    „      „         -  306      , 
=  2  •  2  —  3,  or =  1. 

-  17  •  2  •  9  -  306 


62  *  ELEMENTS   OF   ALGEBRA. 


EXERCISES.    XXIX. 

Perform   the   divisions   indicated;    check   by   assigning 
arbitrary  values. 

27  x^y  -  27  V  121m%»-110mV 

^-  -21xy        '  '  -llmhi' 

x^  +  Sx^y  +  3xh/  +  xy^  -  S  a'b  - 12  a'P  +  9  a^b^ 

^-  X  '  ^  -3a'b 

a^^3a^  +  3  a^b''  -  7  ab^ 

5. ~      ~* 

—  a 

34  a%^c  -  17  ab'^c^  +  51  a^b^c 
^'  liable 

200  xY  -  "^^  ^V  + 1^5  ^V. 

'^'  25icV 

^'  -  13  ic'"^/' 

5j9^  -  15 j9^g  +  lOj^'g'  -  20^^ 

2(a  +  bY-3(a  +  by  +  2(a  +  bf 

(a  +  by 

48  a;^V  -  36  x^hj  +  72  x^^  -  108  xy 
^^'  12  xy 

(x'  -{-2xy  +  yy  +{x^  +  2xy-^  y'^y 
-(x''Jr2xy  +  y') 

(2x-iy  -{■^{2x-iy-{2x-iy 
^^'  -(2x-iy 

-  52  a^%'  -  78  a'b^'  -  26  a^'^b^^  -  130  a'b^ 
^^'  -  26  a%^ 


DIVISION.  63 


III.    DIVISION  OF  A  POLYNOMIAL  BY  A  POLYNOMIAL. 

88.  As  a  preliminary  to  the  explanation  of  this  foriu  of 
division  it  is  necessary  to  observe  the  following  important 
points : 

1.  In  division,  if  dividend,  divisor,  and  quotient  are 
arranged  according  to  the  descending  powers  of  some  letter, 
then  the  first  term  of  the  quotient  is  the  quotient  of  the 
first  terms  of  dividend  and  divisor. 

That  is,  in  dividing  x^  +  Sx^y  +  3  xy^  +  y^hy  x  +  y,  the  first  term 
of  the  quotient  is  x^.  For  it  has  been  shown  (§  66)  that  the  term  of 
highest  degree  in  any  letter  in  the  product  (dividend)  equals  the  prod- 
uct of  the  terms  of  highest  degree  in  that  letter  in  the  multiplicand 
(divisor  or  quotient)  and  multiplier  (quotient  or  divisor). 

E.g.,  in  dividing  x"^  +  x^y  +  2 x^y'^  —  2xhj^  —  xy^  —  y"^  hy  x*  +  2x^y 
+  3  x2?/2  4-  2  xy^  +  y^,  the  first  term  of  the  quotient  is  x^.  If  the  terms 
in  each  polynomial  were  written  in  reverse  order,  the  first  term  of  the 
quotient  would  evidently  be  —  y^. 

2.  If  the  product  of  the  divisor  and  the  first  term  of  the 
quotient  is  subtracted  from  the  dividend,  a  partial  dividend 
is  obtained  which  is  the  product  of  the  divisor  by  the  other 
terms  of  the  quotient. 

That  is,  in  dividing  x^  +  Sx^y  +  Sxy'^  +  y^hj  x  +  y,we  know  (by  1) 
that  x'^  is  the  first  term  of  the  quotient.     Now  if 

from                                x^  +  Sx^y  +  3  xij^  +  y^ 
we  take  x^  (x  +  y)  or   x^  +    x^y 


the  remainder       .  2  x-y  +  3  xy^  +  y^ 

is  a  partial  dividend  and  is  the  product  of  the  divisor,  x  -^y,  by  the 
other  terms  which  follow  in  the  quotient. 

This  is  evident  because  the  whole  dividend  is  the  product  of  x-{-  y 
by  the  quotient ;  hence,  the  2  x^y  +  3  xy"^  +  y^  is  the  product  oi  x  +  y 
by  the  other  terms  of  the  quotient. 

It  will  be  noticed  that  this  is  similar  to  the  division  with  which  the 
student  has  become  familiar  in  arithmetic  ;  each  remainder  is  the  prod- 
uct of  the  divisor  and  the  rest  of  the  quotient. 


64  ELEMENTS   OF  ALGEBEA. 

89.  The  operation  of  division  can  now  be  explained.  Let 
it  be  required  to  divide  3  x^y  -\-  if  -\-  x^  -^  ?>  xif  by  y  -\- x. 
It  has  been  shown  (§  88)  that,  if  the  expressions  are 
arranged  according  to  the  descending  powers  of  x,  the 
first  term  of  the  quotient  is  x^. 

x^  -^  2xy  -{-    y^  =  quotient. 

Divisor         =  x  +  y  )  x^  +  Sx^y  +  Sxy^  +  y^  =  dividend. 

If  x^  {x  +  y)  or  x^  +     x^y is  subtracted, 

the  remainder  2  x^y  +  3  xy^  -\-  y^  is  a,  partial  dividend, 

tlie  prodiict  of  x  -\-  y 
by  the  rest  of  the  quo- 
tient. . •.  the  next  term 
of  the  quotient  is  2  xy. 
Subtracting  2xy{x  +  y)  or  2  x'^y  4-  2  xy^ 
the  remainder  xy^  +  i/^  is  also  a  partial  divi- 

dend, the  product  of 
«  4-  y  by  the  rest  of 
the  quotient.  .-.  the 
next  term  of  the  quo- 
tient is  2/'^. 
Subtracting  y"^  (x  +  y)  or  xy"-  +  y^ 

there  is  no  remainder,  and  the  division  is  complete. 

90.  Exact  division.  If  one  of  the  partial  dividends  be- 
comes identically  0,  the  division  is  said  to  be  exact.  If  not, 
the  degree  of  some  partial  dividend  will  be  less  than  that  of 
the  divisor ;  such  a  partial  dividend  is  called  the  remainder. 

This  subject  will  be  further  considered  in  the  chapter  on  fractions. 

If  D  ^  dividend,  d  =  divisor,  q  =  quotient,  and  r  =  remainder,  then 
D  —  r  =  dq  ; 
that  is,  if  the  remainder  v^^ere  subtracted  from  the  dividend  the  result 
would  be  the  product  of  the  quotient  and  the  divisor. 

91.  Checks.  1.  Since  the  dividend  is  the  product  of  the 
quotient  and  the  divisor,  one  check  is  by  multiplication. 

.•  D  —  r  =  dq,  any  remainder  should  first  be  subtracted. 
2.  The  work  may  be  checked  by  arbitrary  values. 


DIVISION.  65 

92.    Arrangement  of  work  in  division.     The  full  form  of 
the  work  is  as  follows : 

x^  +  2xy  +2y^  =  quotient. 

Divisor  =  x  +  y) x^  -^  S x^y  +  4:xy^  -i-  5 y^  =  dividend. 

x3  -I-     x^y =x'^{x  +  y). 

2  x^y  +  4  xy'^  +  5y^  =  1st  partial  dividend. 
2x^y  +  2  xy^  =2xy{x  +  y). 

2xy^  +  5y^  —  2d  partial  dividend. 
2xy2  +  2y3  =  2  ys  (x  +  y). 
(See  check  below.)  Sy^  =  remainder. 

It  is  better  in  practice  to  abridge  this  work  as  follows : 

x2  +  2  x?/  +  2  ?/2 
X  +  2/ )  x^  +  3  x2y  +  4  xy2  ^  5  ^3 
x«  +     x2y 

2x^y 

2  x2y  +  2  xy2 


2  X2/2  +  5  y^ 

2  xi/2  -f  2  y^ 

Sy^ 

It  is  still  better  to  detach  the  coefficients  if  possible. 

1  +  2  +  2 
1  +  1)1+3  +  4  +  5 

1  +  1  Check.     Let  X  =  y  =  1. 

(1  +  1)  (1  +  2  + 2)  =  1  +  3  +  4  +  5 
or  2  •  5  =  10. 


2 
2  +  2 


2  +  5 
2  +  2 

3  x^  +  2xy  +  2y'^,  and  3  y^  remainder. 

Similarly,  to  divide  x^  —  1  by  x  —  1. 

1  +  1  +  1 
1-1)1+0  +  0-1 

1  —  1  Check.     Let  x  =  2. 

1  (2  -  1)  (8  -  1)  =  4  +  2  +  1 

^-^  or  1  .  7  =  7. 

1 

1-1  X2  +  X  +  1. 


66  ELEMENTS   OF   ALGEBRA. 

EXERCISES.    XXX. 

Perform  the  divisions  indicated  in  exs.  1-14.  Check 
the  results  by  substituting  such  arbitrary  values  as  shall 
not  make  the  divisor  zero. 

1.  x^  —  y^  loy  X  —  y. 

2.  x^^  —  a^'^  by  x^  —  a^. 

3.  32  a^ -7/  by  2  a  -  b. 

4.  x^  +  x'^y^  +  y^  by  x^  -\-  xy  -\-  y^. 

5.  a^  -{- h^  -\- e^  -  3  abc  hy  a -\- b  +  c. 

6.  a^  +  3a^  +  3a-\-l  hy  a^  +  2a  +  l. 

7.  x^  —  3x^  -{-3x  -^y^  —  1  by  x  +  y  —  1. 

8.  x'^  —  2  ax^  +  2  rt^x  —  a^  by  x'^  —  a^. 

9.  1  by  1  —  ic,  carrying  the  quotient  to  6  terms. 

10.  -a^-2a''  +  2a^  +  6a''  +  a~l  by   -  a^  +  a  +  1. 

11.  _(t6  4-8aS6-14a4Z>2  4.Q,3^3  +  6a2^,4by  a3-3«^Z'  +  2a^'l 

12.  a^-5a^b  +  10a^b^-10a'b^-\-5ab^-b^hya^-2ab-\-b^ 

13.  a;^  +  2/^  4-  «^  —  3  0??/^  by  x^  -\-  y^  -\-  z'^  —  xy  —  yz  —  zx. 

14.  a3*4-xy +  y^  by  ?/^  — cc?/  +  cc^.    (Rearrange  the  divisor.) 

Perform  the  divisions  indicated  in  exs.  15-31  by  using 
detached  coefficients,  checking  as  above. 

15.  £cs  -  5  £c2  _  3000  by  x-h. 

16.  16x^-81?/*  by  2x4-3?/. 

17.  3  ic^  -  7  X  -  2  -  2  £t;2  by  1  4- if;. 

18.  a*  4-  24  a  +  55  by  fi^2  -  4  tt  4-  11. 

19.  x^  -2  a^x^  +  a*  by  x""  -  2  ax  ^  a\ 

20.  x«-3x«  +  6x^-7.t2  4-3  by  x^-2.t2  +  1. 

21.  p^  +jj^  4-  4y^  -9^9  +  3  by  p^  -\- p'^  -  3^  4-  1- 

22.  x^  —  ic^  +  2  x^  4-  4  x^  —  7  ^2  4-  4  a?  —  1  hy  x^  +  x  —  1. 


DIVISION.  67 

23.  x^  +  7  xhf  —  5  xV  —  ic V  +  2  ?/^  —  4  a-y''  by  {x  —  yf. 

24.  26 rx2 _^ 4 ^3 _ 3  ^4 ^ ^5  _  92 ^_^ 55  ^^  «^-3a4-ll. 

25.  24  7M*-14m3-9m2-84  +  43m  by  7  -  3 /?^  +  4  7/2,2. 

26.  ic«-3a;^-5cc5  +  2£c^  +  5a;^  +  4x2  +  2  by  .t«  +  2x-1. 

27.  3  m«  +  7  m^  -  12  «^^  +  2  m^  -  3  wt^  _|_  13  ^^^  -  6  by  m^ 
+  3  m  -  2. 

28.  a;^-x^-2x^  +  5iz;^-5iz;^  +  8x2  +  6ic-12  by  £c« 
-2iz;2  +  3. 

29.  :r«  +  2  a;^  +  3  ic«  +  4  (cc^  +  1)  +  5  iz;*  +  6  ic^  +  7  ic^  +  8  ic 
by  (X  +  V)\ 

30.  10  m«  -  11  iw'  -  3  m^  +  20  wt^  +  10 1)1"  +  2  by  5  m^ 
-  3  ??i2  +  2  m  -  2. 

31.  x^  +  2a;«  +  3a;^4-3ic^  +  3x8  +  3x2  +  2a34-l  by  x^ 
+  x^  +  x^  +  x^  +  X  +  1. 

32.  Divide  the  product  of  {x  -V){x-  2){x  -  3)(x  -  4) 
by  2(4-3x)  +  x2. 

33.  Divide  q^  -{-1  by  2'  +  1,  and  hence  tell  the  quotient 
of  100001  by  11  {q  =  10). 

34.  Divide  4:^^ -{- 2t^  +  5t^  +  St  +  1  by  2^  +  1,  and  hence 
tell  the  quotient  of  42581  by  11. 

35.  Divide  the  sum  of  |  x^  +  4  x*  +  7^  x^  +  11  x'^  +  7  x 
+  4  and  |  x^  +  4  x^  +  6^  x^  +  9  x^  +  4  x  by  their  difference. 

36.  Divide  1  +  x^  by  1  -{-  x  carrying  the  quotient  to  5 
terms.     From  the  form  of  this  quotient  tell  what  the  next 

5  terms  will  be. 

37.  The  product  of  two  polynomials  is  2w*  —  13  m^n  + 
31  m%^  —  38  nm^  +  24  n^.     If  one  of  them  is  m^  —  5  mn  + 

6  n^,  what  is  the  other  ? 

Where  the  time  allows,  the  work  in  Synthetic  Division 
(Appendix  II)  should  be  taken  at  this  point. 


68  ELEMENTS   OF  ALGEBRA. 


REVIEW  EXERCISES.    XXXI. 


1.  Solve  the  equation  2  —  (3  —  4  —  a?)  =  3. 

2.  Solve  the  equation  —  2  ic  +  4  =  —  12.     Check. 

3.  Solve  the  equation  f£c  +  4  =  -Jaj  +  4|-.     Check. 

4.  What  are  the  advantages  in  detaching  the  coefficients 
when  practicable  ? 

5.  What  is  the  etymological  meaning  of  quotient  ?  of 
coefficient  ?    of  associative  ? 

6.  The  cube  of  a  certain  number,  subtracted  from  1, 
equals  9.      Find  the  number. 

7.  What  is  the  sign  of  the  product  of  an  odd  number  of 
negative  numbers  ?     Prove  it. 

8.  If  from  twice  a  certain  number  we  subtract  7  the 
result  is  15.      Find  the  number. 

9.  Three  times  a  certain  number,  subtracted  from  5, 
equals  —  10.      Find  the  number. 

10.  Why   do   you   arrange   both   dividend   and   divisor 
according  to  the  powers  of  some  letter  ? 

11.  Why  do  you  avoid  using  such  an  arbitrary  value  in 
checking  division  as  shall  make  the  divisor  zero  ? 

12.  If  to  three  times  a  certain  number  we  add  2  the 
result  is  five  times  the  number.      Find  the  number. 

13.  What  is  the  value  of 


a\a  -  h^a''  -  2 cQ)''  -  a  -  b  ^  c)  -\-  b~\-  c\ 
when  ^  =  3,  5  =  1,  c  =  2? 

14.  What  is  the  Index  Law  of  Division  ?  Have  you 
proved  it  for  all  values  of  the  indices  ?  If  not,  for  what 
kinds  of  indices  ? 


CHAPTER   V. 

ELEMENTARY    ALGEBRAIC    FUNCTIONS. 

I.     DEFINITIONS. 

93.  Every  quantity  which  is  regarded  as  depending  upon 
another  for  its  value  is  called  a  function  of  that  other. 

E.g.,  with  a  given  principal  and  rate,  the  interest  depends  upon  the 
time  ;  hence  in  this  case  the  interest  is  called  a  function  of  the  time. 
Similarly,  the  expression  ic^  _  3^  +  21  is  a  function  of  x,  etc. 

94.  A  function  of  x  is  usually  indicated  by  some  such 
symbol  as  f(x),  F{x),  f^{x),  P{x),  ■••. 

Thus,  if  the  expression  x^  —  x  -\-  I  is  being  considered,  it  may  be 
designated  by  /(x),  read  "function  of  x,"  or  simply  "function  x." 

If  some  other  function  of  x,  as  x*  —  x^  +  2  x^  —  x  +  4,  is  also  being 
considered,  it  may  be  distinguished  from  the  first  one  by  designating 
it  by  F{x).  read  "/major  of  x,"  or  "/major  function  x." 

P  (x),  fx  (x),  ■ . .  are  read  "  P  function  ic,"  "/-one  function 
ic,"  •  •  • .  The  Greek  letter  <^  (phi)  is  also  very  often  used  in 
this  connection,  </>  (x)  being  read  "  phi  function  x.^' 

95.  If  f(x)  is  known  in  any  discussion,  /(a)  means  that 
function  with  a  put  in  place  of  x. 


E.g.,  if 

/(x)=x2  +  2x  +  1, 

then 

/(a)  =  a2  +  2a  +  l, 

/(2)  =  22  +  2-2  +  1  = 

and 

/(O)  =  0  +  0+1=1. 

60 

70  lOLEMENTS   OF  ALGEBRA. 

96.  A  quantity  whose  value  is  not  fixed  is  called  a  vari- 
able ;  if  the  value  is  fixed,  it  is  called  a  constant. 

E.g.,  in  the  expression  y'^-\-2y  +  ^,  y  may  have  any  value,  and 
hence  y  is  a  variable.     But  when  it  is  said  that  y  -  2  =  3,  the  value 
of  y  is  fixed,  and  hence  ?/  is  a  constant,  5. 
I 

97.  Every  algebraic  expression  which,  in  its  simplest 
form,  contains  several  variables  is  called  a  function  of  those 
variables. 

E.g. ,  x'^  -^  2xy  -{■  y^  is  a  function  of  x  and  ?/,  and  may  be  designated 
by/(x,  2/),  read  "function  of  x  and  ?/,"  or  simply  "/of  a;  and  y."" 
But  x  +  y  +  a  —  y  —  xis  not  a  function  of  x  and  y. 

EXERCISES.    XXXII. 

1.  If  f(x)  =  x'-x^  +  x-l,  what  are  f(a),  f(a^), /(- 2), 
/(l),/(0)? 

2.  If  f(x)  =  x'^-\-x-{-l,  and  F(x)=x  —  1,  find  the  value 
of /(ic)  •  F(x).  Check  by  letting  x  =^  2,  i.e.,  by  finding  the 
valueof/(2).i^(2). 

3.  If  f(x)  =  x^  +  3x^  -^  3 X  +  1,  and  <ty(x)  =  x""  +  2x +  1, 
find  the  value  of  f(x)  -^  4,  (x).    Check  by  using  /(I)  -^  </>  (1). 

4.  It  f(x,  y)  =  x^ -Sxhj  +  Sxf  -y^,  Sind  fi(x,  y)  =  x-y, 
find  the  value  oif(x,  y)-fi(x,y);  also  oif{x,y)^f^(x,y). 
Check  by  using /(2,  1)  and/i(2,  1). 

5.  If  F{x,  y,z)  =  x^  +  if  +  z^  -3  xyz,  and/(ir,  y,  z)  = 
X  -\-  y  -\-  z,  find  the  value  of  F(x,  y,  z)  -i-f{x,  y,  z).  Check 
by  letting  x  =  y  =  z  =  1. 

6..  li  f^{x)=  x^  +  2x  +  1,  f^(x)=  x"  -2x  +  1,  and/8(a;) 
=  x'^-l,  find  the  value  of  /i  {x)  -f^  (x)  -f^  {x).  Check  by 
letting  cc  =  2. 

7.  If  f{x)  =  a;*  -  10 a;«  +  35  a^2  _  5Q ^  _^  24,  find  the  values 
of/(l),/(2),/(3),/(4). 


I^H98.  An  algebraic  expression  is  said  to  be  rational  with 
respect  to  any  letter  when  it  contains  no  indicated  root  of 
that  letter.  In  the  contrary  case  it  is  said  to  be  irrational 
with  respect  to  that  letter. 

E.g.^  4a  +      v2  is  rational  with  respect  to  a, 
but  2  +  4  \a  is  irrational  witli  respect  to  a. 

So  x2  —  X  Va  +  va  is  an  irrational  function  of  a,  but  it  is  a 
rational  function  of  x. 

99.  A  rational  algebraic  expression  is  said  to  be  integral 
with  respect  to  a  letter  when  this  letter  does  not  appear 
in  any  denominator.  In  the  contrary  case  it  is  said  to  be 
fractional. 

E.g., is  an  integral  algebraic  expression,  with  respect  to  a, 

but         2  —  is  a  fractional  expression,  with  respect  to  a. 


W 


X         1 

x2 1-  —  is  an  integral  function  of  x, 


a      a 


but        x2  —  Vx  is  not,  because  it  is  not  rational, 


1       2 

-is  not,  because  it  has  x  in  both  denominators. 

X      x2 


«lOO.    An  algebraic  expression  is  said  to  be  homogeneous 
en  all  of  its  terms  are  of  the  same  degree. 
E.g.,  7  a^x  +  4  a^  +  x'^  is  homogeneous,  but  3  a^x  +  4  ax^  is  not. 
So  ax'^y  +  Jfixy'^  +  chj^  is  homogeneous  as  to  x  and  y,  but  not  as  to 
^alone,  nor  as  to  y  alone,  nor  as  to  a,  x,  and  y. 

101.  An  algebraic  expression  is  said  to  be  symmetric  with 
respect  to  certain  letters  when  those  letters  can  be  inter- 
changed without  changing  the  form  of  the  expression. 

E.g.,  x'^-\-2xy  +  y^  is  symmetric  as  to  x  and  y,  because  if  x  and  y 
are  interchanged  it  becomes  y2  ^  2  yx  +  x^  which  is  the  same  as  the 
original  expression.  Similarly,  x^  +  y^  4-  z^  +  axyz  is  symmetric  as 
to  X,  y,  and  z,  but  not  as  to  a,  x,  y,  and  z. 


72  ELEMENTS   OF   ALGEBRA. 

102.  An  algebraic  expression  is  said  to  be  cyclic  with 
respect  to  certain  letters  in  a  given  order  when  its  value 
is  not  changed  by  substituting  the  second  for  the  first,  the 
third  for  the  second,  and  so  on  to  the  first  for  the  last. 

E.g.,  a  (a  —  6)  +  6  (6  —  c)  +  c  (c  —  a)  is  cyclic  as  to  a,  &,  and  c ;  for 
if  6  is  substituted  for  a,  c  for  6,  and  a  for  c,  it  becomes  6  (6  —  c)  + 
c  {c  —  a)  -^  a  {a  —  b),  which  is  the  same  as  the  original  expression. 

It  will  be  noticed  that  if  an  expression  is  symmetric  it 
must  be  cyclic,  for  a  cyclic  change  of  letters  is  a  special 
case  of  the  general  interchange  of  symmetry.  But  the  con- 
verse is  not  true,  for  the  special  case  does  not  include  the 
general  one. 

E.g.,  x'^  +  y^  -\-  z"^  —  x{x  -\-  y'^)  —  y  {y  +  z^)  —  z{z  -\-  x^)  is  cyclic  but 
not  symmetric ;  but  x^  +  y^  +  z'^  —  xy  —  yz  —  zx  is  symmetric  and 
hence  also  cyclic. 

The  theory  of  cyclic  functions  is  often  called  cy do-sym- 
metry, or,  where  no  misunderstanding  will  result,  simply 
symmetry. 

EXERCISES.    XXXIII. 

Select  from  exs.  1-13  those  expressions  that  are  (1)  homo- 
geneous, (2)  symmetric,  (3)  cyclic,  as  to  any  or  all  of  the 
letters  involved : 

1.    a^x  —  b^x  +  c*ic.  2.    ic^  4-  2  x^y^  +  y^. 

3.    x^z  —  3  xyz^  +  y^z^.  4.    ab  -{-  be  -\-  ca  -\-  abc. 

6.    a''  +  b^-]-c^-Sabc. 

6.  abc  -  3  ac^ -^  bc^  —  c\ 

7.  x^  —  x^y  +  ccy  —  xy^  +  ?/*. 

8.  a\b-e)  +  b\c-a)-hc\a-b). 

9.  a^  (b  -  ef  +  b\c-  ay  -\- c\a  -  bf. 

10.  x"^  -^y^  ^z^  +  ax  -\-by  +  ez-^  mxyz. 

11.  be  (b  -\-  c)-\-  ca  (c  +  a)  +  a^>  (a  +  ^)  +  2  abc. 


ALGEBRAIC  FUNCTIONS.  73 

12.  a^  (b^  -  c2)  +  b^  (c^  -  a^)  +  c«  (a^  -  b^). 

13.  {a  +  b)  {a^  +  b^-  G^)  +  {b  +  c)  (b^  +  c^  -  a^) 

+  (c  +  a)  (c^  +  a^-  b^. 

Select  from  exs.  14-20  those  functions  of  x,  of  y,  and  of  a 
that  are  (1)  rational,  (2)  integral  functions  of  those  letters. 

14.  ^x  —  X  wa. 

15.  x^  -\-x^  +  1. 

16.  x'^/a  —  x/a^. 

17.  ic^  +  ic^  +  ic  +  -y/x. 

18.  a;«  +  3  a;^;/ +  3  ic?/2  +  7/8. 

19.  cc"*  +  x"'-^  +  x'»-2  H Y-x"^  +  x  +  1. 

m 

20.  x"*  —  ic^,  (1)  when  m  is  even,  (2)  when  m  is  odd. 


I 


The  applications  of  homogeneity  and  symmetry  are  numerous 
and  valuable.  If  the  time  allows,  they  should  be  taken  at 
this  point.  They  are  set  forth  at  some  length  in  Appendix 
II. 

It  should,  however,  be  said  that  symmetry  and  homo- 
geneity form  two  valuable  checks,  especially  in  multiplica- 
tion. If  two  expressions  are  homogeneous  their  product  is 
evidently  homogeneous. 

E.g.^  the  product  of  x^  +  2x?/  —  y^  and  x  —  y  cannot  be  x^  +  x'^y^ 
3  x?/  +  2/^,  because  the  factors  being  homogeneous  the  product  must 
be  so. 

Likewise,  if  two  expressions  are  symmetric  as  to  two  or 
lore  letters,  their  product  must  be  symmetric  as  to  those 
letters. 

E.g.^  the  product  oix'^  —  2xy  -\-  y^  and  x  -\- y  cannot  be  x^  —  x'^y  + 
+  2/8,  because  this  is  not  symmetric  as  to  x  and  y. 

A  knowledge  of  symmetry  and  homogeneity  is  of  great 
lue  in  factoring. 


74  ELEMENTS  OF  ALGEBRA. 

II.  THE  REMAINDER  THEOREM. 

103.  If  we  consider  the  remainder  arising  from  dividing 
a  function  of  x,  say  x^ -{■  px  +  q,\}j  x  —  a,  we  find  an  inter- 
esting law. 

X  +  p      -\-  a  =  quotient 
X  —  a  )  x2  +  px    +  5 

x^  —  ax 

(p  +  a)  X  +  5 

[p  -\-  a)  X  —  pa  —  a'^ 


a^  +  _pa  +  g  =  remainder. 

That  is,  the  remainder  is  the  same  as  the  dividend  with 
a  substituted  for  x. 

Hence,  if  this  law  is  general,  we  may  find  the  remainder  arising 
from  dividing  x^  +  2  x  —  3  by  x  —  2  by  simply  substituting  2  for  x  in 
the  dividend.     This  gives  2^  +  2  •  2  —  3  =  5,  the  remainder. 

Similarly,  it  is  at  once  seen  that,  if  this  law  is  general,  x^'''  +  2  x^  —  3 
is  exactly  divisible  by  x  —  1.     (Why  ?) 

That  the  law  is  general  is  proved  on  p.  75. 

EXERCISES.    XXXIV. 

Assuming  that  the  remainder  can  always  be  found  as 
above  stated,  find  the  remainders  arising  from  the  follow- 
ing divisions : 

1.  cc"  —  1  by  a?  —  1. 

2.  x'^^  —  y^  by  X  —  y. 

3.  2  cc^  -  64  by  ic  -  2. 

4.  32  a' -1  by  2  a  -  1. 

5.  (7cc)i°  +  l  by  7x  +  l. 

6.  x^  —  x^  -^x^  —  X  +  1  by  X  —  3. 

7.  x^  -{-  x^  —  x^  —  X  +  1  hy  X  —  1. 

8.  3x^  +  4.x^-2x-36  hj  x-2. 

9.  £c"  +  1  by  a;  +  1,  i.e.,  by  x  -  (-  1). 


ALGEBRAIC   FUNCTIONS.  75 

104.  The  Remainder  Theorem.  If  f  (x)  is  a  rational  inte- 
gral algehraic  function  of  x,  then  the  remainder  arising 
from  dividing  f  (x)  hg  x  —  Si  is  f  (a). 

Proof.   1.    Let  q  be  the  quotient  and  r  the  remainder. 

2.  Then  f(x)  =  q(x  —  a)-\-  r.         Def .  of  division 

(I.e.,  the  dividend  equals  the  product  of  the  quotient  and  the  divisor, 
plus  the  remainder,  and  this  is  true  whatever  the  value  of  x.) 

3.  Step  2  is  true  it  x  =  a,  it  being  an  identity. 

4.  But  r  does  not  contain  x.  (Why  ?)- 

5.  .  • .  f(a)  =  q  (a  —  a)  +  r  =  0  -\-  r  =  r,  from  step  3. 

6.  I.e.,  the  remainder  equals /(a),  or  the  dividend 
with  a  substituted  for  x. 

105.  CoROLLAKiES.  1.  If  f  (x)  is  a  rational  integral 
algehraic  function  of  x,  then  the  remainder  arising  from 
dividing  f  (x)  by  x  +  a  is  f  (—  a). 

For  X  -\-  a  =  x  —  {—  a)  ]  hence,  —  a  would  merely  replace  a  in  the 
above  proof. 

[2.  If  f  (a)  =:  0,  then  f  (x)  is  divisible  bg  x  —  a. 
Tor  the  remainder  equals  /  (a) ,  and  this  being  0  the  division  is  exact. 

{.  If  n  is  a  positive  i?iteger. 

(a)  x°  4-  y°  is  divisible  bij  x  +  y  when  n  is  odd. 

Tor,  putting  —  y  for  x,  x»  +  2/"  becomes  ( —  y)"  +  y^^  which  equals 
"when  n  is  odd,  and  not  otherwise. 

L^^      (b)  x°  +  y*^  is  never  divisible  by  x  —  j. 

I^KFor,  putting  y  for  x,  x«  +  2/"  becomes  2/»  +  2/",  which  is  not  0. 

IB        (c)  x"  —  y°  is  divisible  by  x  +  j  when  n  is  even. 

For,  putting  —  y  for  x,  x"  —  ?/"  becomes  (—?/)"  —  y^,  which  equals 
1    0  when  n  is  even,  and  not  otherwise. 

IB       (d)  x"  —  y"^  is  always  divisible  by  x  —  y. 
mK'PoY,  i^uttiug  y  for  x,  x**  —  ?/"  becomes  0. 

B 


76  ELEMENTS   OF   ALGEBKA. 

Illustrative  problems.  1.  Pind  the  remainder  arising  from 
dividing  (x  -\- ly  —  x^  —  1  by  x  +  1. 

Substitute  —  1  for  x,  and  f{x)  becomes  (—  1  +  1)^  —  (—  1)^  —  1, 
which  equals  0  +  1  —  1,  or  0. 

2.  Also  when  (x  —  my  +  (x  —  ny  -\-  (m  -\-  ny  is  divided 
by  ic  +  m. 

Substitute  —  m  for  z,  and  f{x)  becomes  (—  m  —  m)^  -|-  (_  m  —  n)^ 
+  (m  +  n)3,  which  equals  —  8  m^  —  (m  +  n)^  -f  (m  +  n)^,  or  —  8  m^. 

3.  Also  when  nx'"'^^  —  (ti  +  1)  ic"  +  1  is  divided  by  x  —  1. 
Substitute  1  for  x,  and  w  —  (n  +  1)  +  1  =  0. 

4.  Find  the  remainder  arising  from  dividing  x^  -{-  5x* 
-Zx^-2x  +  l  by  x  +  1. 

Here  it  is  rather  tedious  to  substitute  —  7  for  x.  If  the  student 
miderstands  synthetic  division  (Appendix  II)  it  is  better  to  resort  to 
it,  as  follows  : 

11  +  5-3+0-2+        7 
-71-7      14-77      539     -3759 

1-2      11-77      537  ;  -  3752  remainder. 
Check.     [8  -  (-  3752)]  -  8  =  470. 

EXERCISES.     XXXV. 

Find  the  remainders  in  the  following  divisions : 

1.  ic^"'  +  ?/2'»  by  x  +  tj. 

2.  a;^  —  4  £c2  +-  3  by  £c  +  4. 

3.  aj^^'  +  i  +  2/2m  +  i  ^j  x  +  y. 

4.  32a;i0-33a;«  +  l  by  a;  -  1. 

5.  a;*  +  2a;2-3a;-7  by  a;  -  2. 

6.  aj^"^  +  a;io  -  2  by  ic  -  1 ;   by  cc  +- 1. 

7.  i««  +  3  cc2  +  50  by  a;  +  5 ;    by  cc  -  5. 

8.  x^  +  y^  by  a;2  +  y\     (Substitute  -  y"-  for  x\) 

9.  x^^  +  2/15  by  ic^  +-  y^. 

10.      iC^O  +  2/20     l3y     ^4  _|_  ^4_ 


M 


ALGEBRAIC   FUNCTIONS.  77 

REVIEW   EXERCISES.    XXXVI. 

1.  Solve  the  equation  f(x)=f{2). 

2.  If  f(x)  ~x  —  l,  solve  the  equation  f(x)  -/(S)  =  0. 

3.  If  f{x)  =  x  —  1,  solve  the  equation  [/(^)]^=  a;'^  —  3. 

4.  If  F(x)  =  x^  —  5x  -\- 1,  solve  the  equation 

F(x)  =  F(x)  +  5x. 

5.  Is  ax^  +  bxy  +  a?/^  symmetric  as  to  x  and  y?  as  to 
a  and  b?   as  to  a  and  x  ? 

6.  Is  this  a  rational  function  of  x  : 

^x^-x^V^  +  3x-y/a--i^? 
Is  it  an  integral  function  of  a?  ?     Is  it  a  rational  function 
of  a? 

7.  If  f(x,  y)  is  symmetric  as  to  x  and  ?/,  is  [/(ic,  y)]^ 
also   symmetric   as   to   x   and  3/  ?      Illustrate  by  letting 

(a;,  y)  =  x  +  y. 

8.  May  /(ic,  ^)  be  not  symmetric  as  to  x  and  y,  and 
[/(^j  2/)]^  be  symmetric  ?  Illustrate  by  letting  f(x,  y)  = 
x-y. 

9.  Do  you  see  any  advantage  in  having  a  function  sym- 
bol, as  f{x),  in  the  way  of  brevity  ? 

10.  Multiply  x^  -\-3x^y  +  4.  x'^y^  -[-^xtf  +  y^  by  ic^  —  xy 
y'^,  checking  the  result  (1)  by  symmetry,  (2)  by  homo- 
geneity. 

11.  Multiply  x^  —  S  xhj -{■  S  xy^  —  y^  by  x'^  +  2xy-y^ 
d  check  by  symmetry  or  by  homogeneity  according  to 

hich  one  applies. 

12.  Divide  x^  —  y^  hj  x  —  y,  checking  the  quotient  by 
mogeneity. 

13.  Divide  x^  +  y^  hj  x  -\-  y,  checking  the  quotient  by 
mmetry. 


CHAPTER   YI. 
FACTORS. 
I.    TYPES. 

106.  Tlie  factors  of  a  rational  integral  algebraic  expression 
are  the  rational  integral  algebraic  expressions  which  multi- 
plied together  produce  it. 

In  the  expression  3x(x  +  l)(x'^  +  x -\- 1)  (x^  +  2) 

3  is  called  a  numerical  factor, 

X       "        "  monomial  algebraic  factor  of  the  first  degree, 

ic  + 1       "        "  linear  binomial  factor, 

aj2  _j_  a;  -|- 1  "  "  quadratic  trinomial  factor,  the  term  "  quad- 
ratic "  being  applied  to  integral  algebraic  expressions  of  the 
second  degree  in  some  letter  or  letters. 

ic^  -f  2  is  called  a  cubic  binomial  factor,  the  term  "  cubic  " 
being  applied  to  integral  algebraic  expressions  of  the  third 
degree  in  some  letter  or  letters. 

E.g.,  in  the  expression  x^{x  -\-  y  +  z)  {x^  +  y^),  x^  is  a  monomial 
cubic  factor,  x  -\-  y  +  z  is  a.  linear  trinomial  factor,  and  x^  +  y^  is  a. 
quadratic  binomial  factor. 

107.  Rational  integral  algebraic  expressions  which  in- 
volve only  rational  numbers  are  said  to  exist  in  the  domain 
of  rationality. 

JS.gr,,  x2  +  2ic  +  i,  but  not  x2  —  V2.  The  former  has  no  algebraic 
fraction,  and  the  latter  involves  an  irrational  number. 

78 


FACTORS.  79 

108.  The  product  of  two  integral  expressions  in  the 
domain  of  rationality  is  evidently  another  integral  expres- 
sion in  that  domain.  We  say  that  an  expression  is  reducible 
in  the  domain  of  rationality  if  it  is  the  product  of  several 
integral  expressions  in  that  domain,  and  irreducible  in  the 
contrary  case. 

E.g.^  4x2  _  9  is  reducible,  because  it  equals  (2x  +  3)  (2  x  —  3),  but 
x^  —  3  is  not  reducible,  the  word  "  reducible  "  alone  meaning  "  reduci- 
ble in  the  domain  of  rationality. ' ' 

109.  A  rational  integral  algebraic  expression  is  said  to  be 
factored  when  its  irreducible  factors  are  discovered. 

E.g.,  the  factors  of  x*  —  1  are  x^  +  1,  x  +  1,  and  x  —  1.  When 
X*  —  1  is  written  in  the  form  (x^  +  1)  (x  -f  1)  (x  —  1),  it  is  said  to  be 
factored,  because  x2  +  l,x+l,x  —  1  are  irreducible. 

The  expression  x  —  1  is  irreducible,  although  it  has  the  factors 
Vx  +  1  and  Vx  —  1,  because  these  are  not  rational. 

The  term  "  factorable  "  is  applied  only  to  rational  inte- 
gral expressions.  1^-g-,  while  ("V^  4-  l)(v^  —  1)^  "^  —  1? 
expressions  like  Vx  —  1  are  not  spoken  of  as  factorable. 

110.  Factoring  is  the  inverse  of  multiplication,  and  like  all 
inverse  processes  it  'depends  on  a  knowledge  of  the  direct 
process  and  of  certain  type  forms  already  known.- 

E.g..,  because  we  know  that 
Ik  {X  +  ?/)2  =  x2  +  2  x?/  +  2/2, 

r  ifierefore  we  know  that  the  factors  of 

x2  +  2  xy  +  ?/2  are  x  +  y  and  x  +  y, 
and  those  of         m2  +  2  m  +  1      "  m -t- 1     "    m  +  1. 

I^Bll.   Although  all  cases  of  factoring  give  rise  to  identi- 
IBs,  the  symbol  =  is  usually  employed  instead  of  =  as 


80 


ELEMENTS   OF  ALGEBRA. 


112.   The  tjrpe  xy  +  xz,  or  the  case  of  a  monomial  factor. 
Since  x  {y  -\-  z)  =  xy  -\-  xz,  it  follows  that  expressions  in 
the  form  of  xy  +  xz  can  be  factored. 

^.^.,  4a;2  +  2x  =  2x(2x  +  l).         Check.     6  =  2-3. 

A  polynomial  may  often  be  treated  as  a  monomial,  as  in 
the  second  step  of  the  following : 

2/2  —  my  +  ny  —  mn  =  y  {y  —  m)  -\-  n{y  —  m) 
=  iy  +  n){y  -  m). 
Check.     Let  2/  =  2,  m  =  n  =  1.     Then  3  =  3.1. 

It  must  be  remembered  that  an  expression  is  not  factored 
unless  it  is  written  as  a  single  product,  not  as  the  sum  of 
several  products. 

E.g.,  the  preceding  expression  is  not  factored  in  the  first  step  ;  ouly_, 
some  of  its  terms  are  factored. 

EXERCISES.    XXXVII 

Factor  the  following  expressions  : 
1.    ic^  +  x^y  +  ic*. 


3.    x^  —  x^  —  x^-{-  X. 
7.    m^  +  3  m^Ti  +  3  m7i2. 


2.  a'^ -\- 2  ab  -{- 3  ac. 

4.  3x^  —  4.ax^  +  x^. 

6.  ahy  —  ay  -{-  y^  —  hy. 

8.  w'^  —  wy  -\-  ivx  —  wxy. 


113.    The  type  x^  ±  2  xy  +  y^,  or  the  square  of  a  binomialj 
Since  {x  ±yy=x^±2xy  +  y^  (§  69,  1,  2),  it  follows  that 
expressions  in  the  form  oi  x^±2xy  -[-  y'^  can  be  factored. 
E.g.,  a;2  +  4x  +  4  =  (x  +  2)2.  Check.     9  =  32. 

x2  _  6 x?/  +  9 y2  ^  (aj  -  3  vY.         Check.     4  =  (-  2)2. 


EXERCISES.    XXXVIII. 

Factor  the  following  expressions  : 

1.    a;2  +  10cc  +  25.  2.    4. x^  +  4. xy -\- y\ 

3.    25  +  a;2-10a;.  4.    m«  +  14m8  +  49. 


FACTORS.  81 

5.    121ic2-22£c  +  l.  6.    4a;2  4-42/(2/ -2a;). 

7.    9x^-2^xy  +  16y^  8.    81  a;^  +  72 ^y  +  16 3/^ 

9.    4:9z^  +  Slw^-126zw.        10.    (x  +  yy  +  2(x  +  y)-\-l. 

11.  169a^-\-169b^-S3Sab. 

12.  a2^4«^  +  4  +  2(a  +  2)  +  l. 

113.    «2_^2a5  +  ^»2  +  2(a  +  ^)2/  +  2/''. 
14.    x^  -\-2xy  +  y^  +  2xz-j-2yz-{-  z\ 
15.    m^  +  71^  +  p2  +  2  mn  —  2  mp  —  2  np. 
14.    The  type  x^  —  y^,  or  the  difference  of  two  squares, 
ince  {x  +  y)(x-y)  =  x^-  y"-  (§  69,  3),  it  follows  that 
expressions  in  the  form  of  x^  —  y"^  can  be  factored. 

5  •  -  3. 


5-3    -1. 


3   1. 


E.g.,  x'-2-16  =  (x  +  4)(x-4). 

Check. 

-15 

x*  -  16  =  (x2  +  4)  (x2  -  4) 

=  (x2  +  4)  (X  +  2)  (X  -  2). 

Check. 

-15  = 

X4  4-  x22/2  +  y*  =  X*  +  2  x22/2  +  2/^  -  x22/2 

=  (X2  +  2/2)2  _  r^2y2 

=  (X2  +  2/2  4.  a;y)  (a;2  4.  2/2  . 

-xy). 

Check 

EXERCISES.    XXXIX. 

Factor  the  following  expressions  : 
1.    ^^-162/'.  2.    ici6_l. 

3.    a"  +  a^^^  _^  ^>*.  4.    36a;2-92/'. 

5.    16a;*  +  4a;22^2_^^4  6     81  ic*  +  9  a;^  +  1. 

7.   a;2  4-2a;2/  +  2/'-^'.  8.    (cc  +  y)^  -  (i»  -  2/)^- 

9.    a'' +  h'' -  x"" -1+2  ah +  2x. 

I  10.    a2-|-2a^>  +  &2_(a;2_2iC2/  +  2/^). 

n  11.    4a2-f-4a-3(=4a2_^4a  +  l-4). 


82  ELEMENTS   OF  ALGEBRA. 

115.  Forms  of  the  factors.  Although  a  rational  integral 
algebraic  expression  admits  of  only  one  distinct  set  of 
irreducible  factors,  the  forms  of  these  factors  may  often 
appear  to  differ. 

E.g.,  since     {x  -  2y)  {2x  -  y)  =  2x'^  -  5xy  -]-  2y^, 
and  {2y -x){y-2x)  =  2x^-5xy  +  2y2, 

it  might  seem  that  2x'^  —  5xy  +  2y^  has  two  distinct  pairs  of  factors. 

This  arises  from  the  fact  that  the  second  pair  is  the  same  as  the 
first,  except  that  the  signs  are  changed,  each  factor  having  been  multi- 
plied by  —  1.  But  this  merely  multiplies  the  whole  expression  by 
-  1  •  -  1,  that  is,  by  +  I. 

Hence,  the  signs  of  any  even  number  of  factors  may  he 
changed  without  changing  the  product. 

E.g.,  jc2  -  5 X  +  6  =  (x  -  2)  (x  -  3),  or  (2  -  x)  (3  -  x). 
Check.     2  =  -  1  •  -  2,  or  1  •  2. 

X*  -  1  =  (x2  +  1)  (X  +  1)  (X  -  1) 

zz:(x2+l)(-X-l)(l-X) 
=  (-x2-l)(X+l)(l-x). 

Check.    •Letx  =  2.     Then 

16-l=5.31  =  5-3--l=-5-3-  -1. 

EXERCISES.    XL. 

Factor  the  following,  giving  the  various  forms  of  the 
results  and  checking  each. 

1.    l-a\  2.    x^-1. 

3.    16 -£C*.  4.    a^  —  h\ 

5.    lQx^-^ly\  6.    2  +  ^* -2^2^. 

7.    121-f  £c2-22£c.  8.    z^  +  2-2z^V2. 

9.    a;i<>-26a;«  +  168.  10.    a'' -  c'' -^r  b""  +  2  ah. 

11.    16£c*-f-8a;2  +  i_252/^      12.    -x''-l^x^-{-2^x^-lx. 

13.  121  x^  -f  121 2/2  -  9  -  242  xy. 

14.  4.x''-\-l-t/-2ijz-z^  +  4.x. 


FACTORS. 


83 


116.  The  type  x^  ±  3  x^y  -f-  3  xy^  ±  y^,  or  the  cube  of  a 
binomial. 

Since  (x  ±tjy  =  x^±3  xhj  +  3xi/±  if  (§  69,  4,  5),  it 
follows  that  expressions  in  the  form  of  ic^  ±  3  x^y  +  3  xif 
±  y^  can  be  factored. 

E.g.,  8x3  +  12x2 +  6x  +  1  =  (2x)3  +  3(2x)2  +  3  •  2x  +  1 

=  (2  X  +  1)3.  Cheok.     27  =  33. 

27  x6  -  54  x%  +  36  xV  -8^/3  = 

(3x2)3  -  3  (3x2)2  .  27/  +  3  •  3x2(2 2/)2  -  (2?/)3 

=  (3x2-2?/)3.  C/iecA;.     1  =  18. 

x8    x2?/2    xy*    y«     /x\3    o{^V{y'^\  .o(^\{y'^\^    /v'^V 

S~'~T^~6r-Y7  =  \2)~^\2)\'3)'^^\2)\B)   ~\s) 

_/X_^\3 

~  V2       3/ 

Check.     Let  X  =  2,  y  =  3.     Then  l_9  +  27-27=-8  =  (l-  3)3. 


EXERCISES.    XLI. 

Factor  the  following  expressions  : 
1.    l-Sx  +  Sx'^-x^  2.    a^-3a^  +  3a-l. 

x^-^- 3x^  +  3x^-1.  4.    27x3-27a;2  +  9ic -1; 

6.    a^-3a%^  +  3a^b'-b\ 

6.  27  0^9  -  27  a«  +  9  a^  -  1. 

7.  8  a;^  -  12  £c2?/ +  6  a;y2  _  ^3^ 

8.  54.x^-27x  +  Sx\-36x^ 

9.  1.331  ic^  -  7.26  ^2  _^  6.6  X- 8. 

10.  64  xY  -  48  xy  +  12  x'Y  -  1. 

11.  xY^^  +  6  xy^2  _^  ;l2  a;y«  +  8. 

12.  0.125  x«- 0.75  x*  + 0.15  cc^-l. 

13.  (a  +  ^')«  +  3(a  +  ^-)2  +  3(a  +  ^>)H-l. 


84  ELEMENTS  OF  ALGEBRA. 

117.  The  type  x°  ±  y".  It  has  been  shown  (§  105,  Re- 
mainder Theorem,  cor.  3)  that 

£c"  _|_  y»  contains  the  factor  x  -{■  y  when  n  is  odd, 
"  "  "        ^^      X  —  y  never, 

^«.  _  yn        u  a       u      rf^  j^  y  ^Jien  n  is  even, 

"  "  "        "      cc  —  ?/  always. 

Hence,  it  follows  that  expressions  in  the  form  of  x""  ±  y^ 
can  often  be  factored. 

E.g.^  x^  +  y^  contains  the  factor  x  ■\-y.  The  other  factor  can  be 
determined  by  division.  It  may  also  be  determined  by  noticing  that 
a;3  ^  yZ  ig  symmetric  and  homogeneous,  and  that  its  factors  must 
therefore  be  x  +  ?/  and  x^  _|.  ]^xy  +  ?/2,  where  k  is  to  be  determined. 
Letting  x  =  y  =  1, 

x3  +  2/3  =  (X  +  y)  (x2  +  kxy  +  ?/2) 
becomes  2  =2(2  + A;), 

and  therefore,  A;  =  —  1, 

whence  x^  +  t/^  =  (^c  +  2/)  (a?^  —  a??/  +  y^). 

This  type  occurs  so  often  that  the  forms  of  the  quotients 
should  be  memorized : 


signs  alternating. 


the 


03"  —  y- 


signs  alternating. 

3.  ~^  =  cc^-i  +  a;"-2y  +  a;"-y  +  x^^-^f  +  •  •  •,  the 
signs  being  all  +. 

We  are  thus  able  to  write  out  the  quotient  of  (xi^  +  ?/i5)  ^  (x  +  y) 
at  sight,  and  so  for  other  similar  cases. 

The  integral  parts  of  the  quotients  in  1  and  2  are  the  same,  but 
the  remainders  are  different.  E.g.^  if  n  is  odd  there  is  no  remainder 
in  1,  but  in  2  there  is  a  remainder  —  2  ?/». 


FACTORS.  85 

When  the  exponent  n  exceeds  3  it  is  better  to  separate 
into  two  factors  as  nearly  of  the  same  degree  as  possible, 
and  then  to  factor  each  separately. 

E.g.,  x8  -  ?/8  =  {x^  +  2/*)  (x*  -  y^) 

=  {X^  +  2/*)  (X2  +  2/2)  (X2  -  2/2) 

=  (X*  +  y^)  {x^  +  y^)  (X  -\-y){x-  y), 

or  the  same  with  certain  signs  changed  (§  115). 

This  is  better  than  to  take  out  the  linear  binomial  x  -\-  y  ov  x  —  y 
first,  which  would  give 

x8  —  2/^  =  (x  +  y)  (ic^  —  x^y  +  x^y"^  —  x^y^  +  x^y^  —  x^y^  +  xy^  —  y"^), 
or  (x  —  y)  (x^  +  x^y  +  x^y^  +  ic*2/^  +  x^y^  +  x^y^  +  xi^  +  y^), 

in  which  cases  it  would  be  difficult  to  discover  the  factors  of  the  two 
expressions  of  the  seventh  degree. 

So  x2«  -  2/2«  =  (x«  +  y^)  (x«  -  ?/"). 

118.  Binomials  of  the  form  cc"  ±  ?/"  which  have  not  the 
factor  x±y  may  contain  ic"*  ±  t/"*. 

X6  +  y6  ^  (X2)3  +  (2/2)8  =  (a;2  +  ?/2)  (x^  -  X'^y'^  +  2/4). 

EXERCISES.    XLII. 

'actor  the  following  expressions  : 


■^t. 

x'  +  1. 

2. 

x'  -  16. 

3. 

x^  -  if. 

4. 

1  -  x^\ 

5. 

x^^%y\ 

6. 

x^  +  f. 

li 

32a^s  +  l. 

8. 

£C2-+1  +  1. 

i 

a;i2  _|_  4096. 

10. 

^2^,4  _  ^,2^4 

'^1 

729^36  +  2/'. 

12. 

216a«-Z»«. 

13. 

{x  +  2/)«  +  1. 

14. 

125a«  +  27. 

15. 

64cc«-7292/^    . 

16. 

27a»  +  64^«. 

17. 

125^^-27cc/. 

18. 

a^  +  a  +  b^  +  b.   . 

19. 

{a  -  by  -(a  +  by. 

20. 

m^-n^  +  2n-l. 

86  ELEMENTS   OF  ALGEBRA. 

119.  Thetypex^  +  ax  +  b.  'Letx^-\-ax  +  b  =  (x-\-m)(x-\-n), 
in  which  m  and  n  are  to  be  determined.     Then 

x^  -\-  ax  +  b  =  x^  -\- (m  -\-  n)  X  -{-  mn. 

It  therefore  appears  that  if  two  numbers,  m  and  n,  can  be 
found  such  that  their  sum,  m  +  n,  is  a,  and  their  product, 
mriy  is  h,  the  expression  can  be  factored. 

E.g.,  consider  a:2  +  lOx  +  21. 

Here  10  =  3  +  7, 

and  21  =  3  •  7, 

x2  +  10  X  +  21  =  (x  +  3)  {X  +  7).     Check.     32  =  4  •  8. 

Consider  also  x^  —  3  x  —  40. 

Here  -3  =  5-8, 

and  -  40  =  5  •  -  8, 

x2-3x-40  =  (x  +  5)(x-8).     Check.      -42  =  6.-7. 

EXERCISES.    XLIII. 

Factor  the  following  expressions  : 

1.  x^  +  Zx  +  2.  2.  x''-x-2. 

3.  ic^  +  a;2_12.  4.  x^-^x  +  Q. 

5.  cc2-4iz;-165.  6.  j^^-j^-GOO. 

7.  a2-aa-130.  8.  a;2_4^_21. 

9.  ^2- 11  a -60.  10.  x'-4.x^-4.5. 

11.  4a;2_|_8£c-45.  12.  a^  + 17  a  +  66. 

13.  ic2  +  41  £c  +  420.  14.  ic^  +  16a;2  +  55. 

15.  a2_24a  +  135.  16.  icy  +  4a;V  +  3. 

17.  x^  -  15  cc2  _  100.  18.  ^2  -  16  a  -  225. 

19.  a'^x^  +  5  a^x""  +  6.  20.  ic^  ^  7  a??/  +  10 1/. 

21.  4a2  +  2aZ.-2&2.  22.  a'x""  -  5  a^x  -  U. 

23.  m2- 38  m  4- 165.  24.  cc^  +  lla^y  -  26/. 

25.  mV  _  7  ^ic  -  18.  26.  m^ic*  +  12  m,x^  +  35. 


FACTORS.  87 


'^"  ax^  -^  bx  -\-  G=  (mx  +  71)  {px  +  q), 

in  which  m,  71,  p,  and  q  are  to  be  determined.     Then 
ax^  -\-hx  -{-  c^i  mpx^  +  (viq  +  pn)  x  +  qn. 

It  therefore  appears  that  the  coefficient  of  x,  mq  +  pn,  is 
the  sum  of  two  numbers  ivhose  product,  mqpn,  is  the  product 
of  the  coefficient  of  x^,  mp,  and  the  last  term,  qn.  Hence, 
if  these  numbers  can  be  detected,  the  expression  can  be 
factored. 
E.g. ,  consider  6  x2  +  17  x  +  12. 
Here  17  =  9  +  8, 

and  6  •  12  =  72  =  9  •  8. 

6ic2  + 17x  + 12  =  6x2  +  9x +  8x  + 12 

=  3x(2x  +  3) +  4(2x  +  3) 
=  (3x  +  4)(2x  +  3).     Check.     .35  =  7-5. 
Consider  also  8  x^  +  7  x  —  3. 
Here  7  =  9-2, 

and  6    -  3  =  -  18  =  9  •  -  2. 

6x2  4-7x-3  =  6x2  +  9x-2x-3 

'  =3x(2x  +  3)  -(2x  +  3) 
=  (3  X  -  1)  (2  X  -f-  3).     Check.     10  =  2  •  5. 

EXERCISES.    XLIV. 

factor  the  following  expressions  : 


6,^2  +  ^-12. 

2. 

12P--P-1, 

4a;2_4cc-3. 

4. 

3  ^2  +  8  a  +  4. 

600  a'^  —  a-  1. 

6. 

9cc2-17cc-2. 

Ux^-5x-l. 

8. 

Sa^  +  22a-{- 12. 

12p''  -  7p  +  1. 

10. 

6^  +  25  0^  +  1). 

1    11 

^12  _  7  ^6^6  _  S  ^12_ 

12. 

16x''-62x  +  2T. 

16  a^  +  43  a^*  +  27  b\ 

14. 

4.0  a^-{- 61  ab-S4:b\ 

15 

16xyz^-{-S9xi/z-27. 

16. 

30x^-4:1  xz-15z\ 

88  ELEMENTS   OF  ALGEBRA. 

121.  Application  of  the  Remainder  Theorem.  The  presence 
of  a  binomial  factor  is  usually  detected  very  readily  by  the 
use  of  this  theorem  (§  104). 

E.g.,  x^  — 4x4-3  evidently  contains  the  factor  (x  —  1),  and  the 
other  factor,  x^  +  x  —  3,  can  be  found  by  division. 
Similarly,  consider  x^  —  2  x  —  21. 
Trying  x  —  1  we  have 

/(I)  =  1  -  2  -  21  ?i  0  ;      .-.  X  -  1  is  not  a  factor. 
Trying  x  -f  1  we  have 

/(-  1)  =  -  1  +  2  -  21  ;zi  0  ;  .-.  X  +  1  is  not  a  factor. 
Trying     x  —  3  we  have  /(3)  =  0  ;  .-.  x  —  3  is  a  factor. 

If  the  student  understands  Synthetic  Division  (Appendix 
II),  the  test  of  divisibility  is  easily  made  by  that  process, 
thus: 

11    0    -2    -21 
3|        3         9         21 

13         7 ;         0  remainder. 

Hence  the  factors  are  x  —  3  and  x^  +  3x  +  7. 
Check.     -  22  =  -  2  .  11. 

Since  the  factors  of  —  21  are  ±  1  and  qp  21,  ±3  and  =F  7,  the 
number  of  trials  necessary  is  very  limited. 

EXERCISES.    XLV. 

Factor  the  following  expressions : 
1.    x8-19i»-30.  2.    x^-Sx-2. 

3.    m*  —  2  mn^  +  n^.  4.    a^  —  a^  —  a  —  2. 

5.    a^-a-2  -\-2a^  6.    a;«  +  9 x^  +  20 cc  +  12. 

7.    a^-6a^-{-lla-6.  8.    a^  +  8a^ -112  a +  256. 

9.    a^'-a^-lBa-h  12. 

For  those  who  have  studied  symmetry  as  set  forth  in 
Appendix  III,  the  cases  of  factoring  given  in  Appendix  IV 
are  recommended  at  this  point. 


FACTORS. 


89 


MISCELLANEOUS    EXERCISES.    XLVI. 
'122.    General  directions. 

..  First  remove  all  monomial  factors. 

}.  Then  see  if  the  expression  can  be  brought  under  some 
of  the  simple  types  given  on  pp.  81-87.  This  can  probably 
always  be  done  in  cases  of  binomials  and  quadratic  trino- 
mials, and  often  in  other  cases. 

3.  If  unsuccessful  in  this,  the  Eemainder  Theorem  may 
tried,  especially  with  polynomials  of  the  form 

£c"  +  ax'^~^y  +  hx'^~^y^  +  •  •  •. 

4.  Always  check  the  results,  and  be  sure  that  the  factors 
are  irreducible. 


™ 


I.  cc*  +  4. 
^3.  x^  +  y^. 
^5.  x^  —  a;*?/*. 

7.  a;^  +  x2  +  ^. 

9.  a%^c^-a%^G\ 

11.  a^-a^- 110. 

L3.  £C*  -  11  ic2  _^  1. 

15.  6a;2_23x4-20. 

17.  ccy  4-  2  ccy  +  xy. 

L9.  a^-lba%''  +  ^b\ 

II.  ah  At  y'^  —  ay  —  hy. 
13.  £c*  -  8  icy  +  16  ^/. 
\h.  (a  +  by  +  (a-by. 

27.  2/«  + 37/^  +  62/ +  18. 

29.  21  ^2  +  26  aZ>  -  15  h''. 


2.  rz;*  +  4  /. 

4.  1  +  x^  +  cc^ 

6.  ic^  +  2/^  +  icy. 

8.  x''-2xSf  +  7/. 

10.  £C2  (a;2  +  7/2^  +  /. 

12.  ic'' +  ic^^  +  2  icY 

14.  2a;2  +  lla;4-12. 

16.  2/2-^2^2^-1. 

18.  (x  +  2/)^  —  ir'^  —  2/"^. 

20.  ax"^  +  {a  +  b)x  +  b. 

22.  12  x22/2  -  17  a;?/ +  6. 

24.  (x  +  l)2-5cc-29. 

26.  16  cc*  -  28  ic2y2  _|_  ^4^ 

28.  7x3  +  96a;2-103£c. 

30.  a;4-((fc2  4-^,2^x2  +  a262^ 


90  ELEMENTS   OF   ALGEBRA. 

31.  m«w«  +  l.  32.  a^  -^a^b^  +  b^ 

33.  9a;2_igy2  34,  a;2"-lla;"  +  28. 

35.  a^  j^a-2  a».  36.  9  a^"*  -  5  -  4  a"*. 

37.  10a2_3606*.  38.  a2(a2_24)  +  63. 

39.  cc'*"'  +  x^"*  +  1.  40.  a^  —  ac  —  bc  —  b'^. 

41.  ic*  +  «»  +  a;%*.  42.  x"-  +  12  £C2/  +  36  t/I 

43.  a;2  ^  16  ic  +  63.  44.  m^  —  ^^  _  ^2  ^  2  ^m;. 

45.  a;2-14a;  +  49.  46.  (a^  +  1)^  _  (^,2  +  i)3. 

47.  a^  (a^  —  1)  -  ^6.  48.  {x  +  y)^  +  4  (w  +  ^)^ 

49.  6  +  15  a^  _  19  a.  50.  5  aZ>  —  Z'C  +  cc?  —  5  ac?. 

51.  S  —(x  +  y  +  zy.  52.  cc^  +  ?/^  —  4  cc^^  —  4  ic?/^. 

53.  a%-ab^  +  a%  +  ab''. 

54.  a2(^_^i)_52(^_l_l), 

55.  Sxy{x  +  y)  +  x^  +  y\ 

56.  4  ccy  -  (a:2  +  2/2  _  ^y . 

57.  2ic+(a;2-4)?/-2a^y/2_ 

58.  121  a*  -  795  ^2^2  _^  9  Z-^ 

59.  (a-4)2-4(a-4)  +  4. 

60.  (a; -5)2 -8(0; -5)+ 12. 

61.  x^{x  —  2y)  —  y^{ij  —  2x). 

62.  l-(a-Z.)-110(a-J)2. 

63.  10  +  16(a4-&)  +  6(a  +  ^>)2. 

64.  (m  +  ny  +  10  (m  +  w)  +  24. 

65.  2  ic2  -  x^y  +  (2/  -  2)  (a;y  -  a;)^. 

66.  x^  +  y^-  (w^  +  g;2)  -|_  2  (xi/  +  wz). 

67.  (a  +  by  -(a  +  by  -  (a  +  ^')'  +  1- 


FACTORS.  91 

II.     APPLICATION   OF  FACTORING  TO   THE  SOLUTION   OF 
EQUATIONS. 

123.  To  solve  an  equation  is  to  find  the  value  of  the 
unknown  quantity  which  shall  make  the  first  member  equal 
to  the  second.  Such  a  value  is  said  to  satisfy  the  equation 
(§  17). 

E.g.,  if  cc2  =  4, 

then  x2  -  4  =  0,  or  (x  +  2)  (x  -  2)  =  0  ; 

.-.  X  =  +  2  or  —  2.  That  is,  either  +  2  or  —  2  will  satisfy  the  equa- 
tion ;  for  if  X  ==  +  2,  then  (2  +  2)  (2  -  2)  ==  0  ;  and  if  x  =  -  2,  then 
(-2  +  2)(-2  -2)  =  0. 

If  x2  +  X  :=  6, 

then  x2  +  X  -  6  =  0, 

whence  (x  +  3)  (x  —  2)  =  0.  This  equation  is  evidently  satisfied  if 
either  factor  of  the  first  member  is  0.     (Why  ?) 

If  X  +  3  =  0,  then  x  =  —  3,  because  —3  +  3  =  0; 

and  if         x  -  2  =  0,      "     x  =  2,  "  2-2  =  0. 

If  x4  -  6x3  + 11x2 -6x  =  0, 

then  X  (x  —  1)  (x  —  2)  (x  —  3)  =  0.    This  equation  is  evidently  satisfied 

any  factor  of  the  first  member  equals  0.     (Why  ?) 

Hence,  x  may  equalO,  as  one  value  ; 

if  X  —  1  =  0,  then  x  =  1,  because  1  —  1  =  0; 

Id  if  x-2  =  0,      "      x  =  2,        "         2-2  =  0; 

id  if  X  -  3  =  0,      "     X  =  3,       "        3-3  =  0. 

EXERCISES.    XLVII. 

Solve  the  following  equations  : 

1.  a;2-l  =  0.  2.  ic2  +  287  =48a;. 

3.  2x^  +  2  =  Bx.  4.  6a;2-13x  +  6  =  0. 

5.  cc2  =  2x  +  143.  6.  x^-10x^  +  21=0. 

7.  x^  -{-'ix''  +  x  =  Q>.  8.  ic6  -  14^4.^49 ^2^  3g_ 

9.  ic^-13£c2  +  36  =  0.  10.  2ic3-67cc2  +  371rr  =  0. 

11.  2x^-lx''  +  ^x  =  0.  12.  .x^-15a;2  +  10j;  +  24=:0. 


92  ELEMENTS   OF  ALGEBRA. 

III.     EVOLUTION. 

124.  If  an  algebraic  expression  is  the  product  of  two 
equal  factors,  one  of  those  factors  is  called  the  square  root 
of  the  expression.  Similarly,  one  of  three  equal  factors  is 
called  the  cube  root,  one  of  four  equal  factors  the  4th  root,  •  •  • 
one  of  n  equal  factors  the  nth  root. 

The  broader  meaning  of  the  word  root  is  discussed  later  (§  130). 

The  process  of  finding  a  root  of  an  algebraic  quantity  is 
called  evolution. 

Evolution  is,  therefore,  a  particular  case  of  factoring. 
It  is  evidently  the  inverse  of  Involution,  as  Eoot  is  the 
inverse  of  Power. 

125.  Symbolism.  Square  root  is  indicated  either  by  the 
fractional  exponent  ^  or  by  the  old  radical  sign  V~,  a  form 
of  the  letter  r,  the  initial  of  the  Latin  radix  (root). 

Similarly,         a^  or  V a  means  the  cube  root  of  a, 

and,  in  general,    a"  "    Va       "       "     nth        "        " 

For  present  purposes  it  is  immaterial  which  set  of  symbols  is  used. 
The  student  should,  however,  accustom  himself  to  the  fractional 
exponent,  which,  while  a  little  more  difficult  to  write,  has  many 
advantages  over  the  older  radical  sign  as  will  be  seen  later. 

126.  Law  of  signs.  Since  any  power  of  a  positive  quantity 
is  positive,  but  even  powers  of  a  negative  quantity  are  posi- 
tive while  odd  powers  are  negative  (§  77),  therefore, 

1.  An  even  root  of  a  positive  quantity  is  either  positive  or 


E.g.,  4^  =  ±2,  81*  =  ±3. 

2.  An  odd  root  of  any  quantity  has  the  same  sign  as  the 
quantity  itself. 

E.g.,  8*  =  2,  (-8)*  =-2. 


FACTORS.  93 

3.  An  even  root  of  a  negative  quantity  is  neither  a  posi- 
tive nor  a  negative  quantity. 
E.g.,  V—  1  is  neither  +  1  nor  —  1. 

An  even  root  of  a  negative  quantity  is  said  to  be  imagi- 
nary, and  imaginary  quantities  are  discussed  later  (Chap. 
XIII). 

127.  The  root  of  a  monomial  power  is  easily  found  by 
inspection. 

E.g.,-.-  4a^¥  =  2-2-aa-b'b-b-b, 

V4a2&4^  V(2  ■ab-b)-{2-a-b-b)  =  ±2-a-b-b 

=  ±2  ab\ 

3    y 

Similarly,  v  64  x^y^  =  4  xy'^, 


V32  xi5y3o  =  2  xh 

6  , 


'64x12=  ±2x2. 


EXERCISES.    XL  VIII. 

Simplify  the  following  expressions  : 


1.   V4V^.  2.    ^-  8  ^6^,12. 


3.    ^3^  (a -2  by.  4.   ^16a''"'-s/a''^b''^\ 

5.    ^2a^\^2b^^I?. 


6.   Vl6a;V°^^   V32icV'^''' 

2x/ 2x+l/ 2x+\/ 


9.    V729^W,   V729<^i«^«,   V729^i«^. 


10.   VV«i^3^^  ^^^  being  even ;  m  being  odd. 


94  ELEMENTS   OF  ALGEBRA. 

128.  Roots  extracted  by  inspection.  The  roots  of  the  mono- 
mials given  on  p.  93  were  extracted  by  inspection.  Simi- 
larly, the  square  root  of  a  square  polynomial,  the  cube  root 
of  a  cube  polynomial,  etc.,  can  often  be  found  by  inspection. 

Illustrative  problems.  1.  What  is  the  square  root  of 
ic*  +  4  x^y  +  4  2/^. 

1.  •••  [±  (/  +  w)]2  =p  4-  2/?i  +  w2,  §  82 

2.  and  •.•  this  polynomial  can  be  arranged  in  a  similar  form,  mz. , 

(x2)2  +  2x2(2  2/)  +  (2?/)2, 

3.  .-.  it  is  evidently  the  square  of  ±  («2  -f  2  y). 
Check.     (±3)2  =  1+4  +  4  =  9. 

2.  rind  the  cube  root  of  x^  +  6^;^  +  12  x^  +  Sy^ 

1.  •.•  (/  +  ri)8  =p  +  3/2n  +  3/7i2  +  n^  §  82 

2.  and  •.•  this  polynomial  can  be  arranged  in  a  similar  form,  viz., 

(x2)3  +  3  (x2)2 .  2  y  +  3  a;2  (2  2/)2  +  (2  y)^ 

3.  .-.  it  is  evidently  the  cube  of  x2  +  2  y. 
Check.     33  =  1  +  6  +  12  +  8  =  27. 

3.  Find  the  square  root  of 

a''  +  4.P  +  9 c^  +  4: ab  -  6  ac  -  12  be. 

1-    '■'  [±{x  +  y  +  z)]2  =  x^-\-y^  +  z^  +  2xy  +  2yz  +2zx, 

2.  and  •.•  this  polynomial  can  be  arranged  in  a  similar  form,  viz., 
a2  +  (2  6)2  +  (_  3  c)2  +  2  a (2  6)  +  2  a(-  3  c)  +  2  (2  6)  (-  3  c), 

3.  .-.  it  is  evidently  the  square  of  ±  (a  +  2  6  -  3  c). 
Check.     02  =  1+4  +  9  +  4-6- 12  =  0. 

4.  Find  the  fifth  root  of 

^10  -5a'b-\-  10  a%-'  -  10  a*b^  +  5  a^b^  -  b^ 

1.  •.•  there  are  6  terms,  and  the  polynomial  is  arranged  according 
to  the  powers  of  a  and  6,  it  is  the  5th  power  of  a  binomial  (§  82)  whose 
first  term  is  a^  and  whose  second  term  is  —  6,  if  it  is  a  5th  power. 

2.  But  (a2  -  6)5  equals  the  given  polynomial.     (Expand  it.) 


FACTORS. 


95 


EXERCISES.    XLIX. 

Extract  the  square  roots  of  exs.  1-6. 

1.  ^x'-ix'  +  ^^. 

2.  4:  a -12  Vab  -\-9b. 

3.  Am^  —  12mx^  +  9x\ 

4.  9  a'b'  -  30  a'b'c'  +  25  a'c^'. 

5.  4  m^  +  4  mn  +  12  mp  -\-  n^  -\-  6  np  -{-  9p\ 

6.  4  ic*  -  12  a^V  +  16  ^^y  +  9  xY  -  24  a;?/*  +  16  y\ 

Extract  the  cube  roots  of  exs.  7-12. 

8.  m^  -{-^  m'^n  +  12  m^Tv^  +  8  ?i^ 

9.  8  x8  -  84  cc^y  +  294  xif  -  343  2/». 

10.  8ic^  +  12a;5  +  18ic4  +  13a;3  +  9a;2  +  3^  +  l. 

11.  77^6  _  3  ^^5  _  3  ^4  _^  ;L;l_  ^8  ^  g  ^2  _  12  m  -  8. 

12.  x'  -  12ccs  +  54  cc*  -  112  a;^  +  108  a;^  _  48  a^  +  8. 

Extract  the  fourth  roots  of  exs.  13,  14. 

13.  3^5^  ^'  +  I  ^'2/  +  I  xy  +  ^x7f-^  if  y\ 

14.  16  a;4  -  96  xhj  +  216  a^y  -  216  xf  +  81 2/*. 

■  Extract  the  fifth  roots  of  exs.  15,  16. 

15.  80a;^-80a;^  +  32a;S-40ic2_j_i0a;-l. 

16.  x^'  -  5  ^8y  _^  5  ^Y  - 1  xY  +  A  ^ V'  -  ^V  2/'- 

Extract  the  sixth  roots  of  exs.  17,  18. 
r.    a«  _  12^5  +  60a*  -  160^8  +  240^^  -  192a  +  64. 
a6  _  2  a^Z*  +  I  a'b^  -  fa  a%^  +  ^j  a^''  -^ab'  +  ^^^  b\ 


96  ELEMENTS  OF  ALGEBRA. 

129.  Square  root  by  the  formula  f ^  +  2  f n  +  n^.  The  subject 
is  best  understood  by  following  the  solution  of  a  problem. 

1.  Required  the  square  root  of  4  a?*  —  12  x^y  +  9  y^. 

Let  /  =  the  found  part  of  the  root  at  any  stage  of  the 

operation,  and 

n  =  the  next  term  to  be  found. 

Then  (f+ny=f'  +  2fn  +  n\  §  82 

The  work  may  be  arranged  as  follows  : 

Root    =±{2x''-^y) 

Power  =     4  cc*  —  12  £c  V  +  9  ^2  contains  /^  +  2fn  +  n^ 

f=     ^ 

2/=4£c2                   -Ux^y  +  dy""       "                 2fn  +  n^ 
2f+n  =  4:X^-Sy  -Uxhy  +  dy""      = '^ 

Explanation.  1.  If  a  root  is  arranged  according  to  the  powers  of 
some  letter,  the  square  obtained  by  ordinary  multiplication  will  be  so 
arranged  (§  65). 

2.  .-.  the  square  is  arranged  according  to  the  powers  of  x,  so  that 
the  square  root  of  the  first  term  shall  be  the  first  term  of  the  root. 

3.  •••  4  x*  =  the  square  of  the  first  term,  the  first  term  is  2  x^. 

4.  Subtracting /2,  the  remainder,  —12x^y  +  9?/2,  contains  2/n  +  n^. 

5.  Dividing  2fn{i.e.,  -  12x^y)  by  2 /{i.e.,  4a;2),  w  is  found  to  be 
-3y. 

6.  •.•  /2  =  4  a:*,  and  2/n  +  n^  =  -  12  x'^y  -\- 9  y^,  .-.  the  sum  of  these 
is  the  square  of  ±{2x^  -  Sy). 

Check.     Jjetx  =  y  =  l.     Then  (- 1)2  =  4  -  12  +  9  =  L 

We  might,  after  a  little  practice,  detach  the  coefficients. 
In  the  above  example  it  would  be  necessary  to  remember 
that  the  powers  of  x  decrease  by  two,  while  those  of  y 
increase  by  one. 

:EJ.g.,  4-12  +  9|2-3 

4 

4  -12  +  9  ±(2x''-3y) 

4-3  -12  +  9 


I 


FACTORS.  97 

2.  Kequired  the  square  root  of 

Root    =±(a-62        +2c) 

Power  =  a2_2tt62  +  &4^4o[c_4  52c_|_4c2  contains /2  +  2/w  +  n2 

2/      =2  a  -2a62  +  64+...  u  2/n  +  ri2 

2/+7i-2a-62      -2a62-f54 ^ ^^ 

2/       =2 a-2 62                         4  ac -4 62c+4 c2  contains          2/n  +  n2 
2/+n=2a-2&2-j-2c 4ac-4b2c+4c2        = '^ 

Explanation.     1 .  See  p.  96  for  explanation  down  to  2/=  2  a  -  2  62. 

2.  •.•  /2  =  a2,  and 

2/n  +  n2  =  -  2  a62  +  6*, 
.-.  (/  +  n)2  =  a2  -  2  a62  +  6*,  the  square  of  a  -  62. 

3.  •.•  a  —  62  has  now  been  found,  it  may  be  designated  by/. 

4.  .-.  4ac  —  4  62c  +  4  c2  contains  2/n  +  ^2,  the  square  of  a  —  62 
having  been  subtracted. 

Check.     Let  a  =  6  =  c  =  1.     Then  22  =  1-2+1  +  4-4  +  4  =4. 
Or  let  a  =  1,  6  =  2,  c  =  3. 
Then  (1  -  4  +  6)2  =  32  =  9 

and  1  -  8  +  16  +  12  -  48  +  36  =  9 ; 

d  so  for  any  other  arbitrary  values. 


y^ffU 


130.  Extension  of  the  definition  of  root.  If  an  algebraic 
expression  is  not  the  product  of  r  equal  factors,  it  is  still 
said  to  have  an  rth  root.  In  such  a  case  the  rth  root  to  n 
terms  is  defined  to  be  that  polynomial  of  n  terms  found  by 
proceeding  as  in  the  ordinary  method  of  extracting  the  rth 
root  of  a  perfect  rth  power. 
kl^L  ■^•^•'  ^^^^  square  root  of  1  —  x  to  5  terms  is 
IB  ±  (1  -  ix  -  1x2  -  ^^r,z  _  _|^x*  -...). 

IH  In  the  same  way  we  may  speak  of  the  square  root  of 
ilTOmibers  which  are  not  perfect  squares.     Thus  the  square 
root  of  2  to  two  decimal  places  is  1.41 ;   to  three  decimal 
places,  1.414,  and  so  on.     We  may  also  speak  of  the  cube 


98  ELEMENTS  OF  ALGEBRA. 

EXERCISES.  L. 

Extract  the  square  roots  of  exs.  1-16. 

1.  x^  +  2x^-x  +  i. 

2.  l  +  8a  +  22a2  4-24a»  +  9a^ 

3.  9(a^-iy-12(a^-l)a-^Aa\ 

4.  x^-6x^-{-4.x^  +  9x^-12x+4:. 

5.  x^-2  ax^  +  aV  _  2  to^  +  2  abx^  +  ^>l 

6.  25a^  +  9b^  +  c^  +  6bc-10ca~S0ab. 

7.  10:c*- 10x8- 12^5 +  5ic2  +  9cc«-2cc  +  l. 

8.  9  ic«  -  12  aa^^  +  4  a^x^  +  6  a^x^  -  4  a*cc*  +  a^x^. 

9.  16  -  Sr/i  -  237?^2  +  227^1^  +  5m*  -  12m^  +  4m^ 

10.  9  a^^*  - 12  a'^b'  +  4  a%'  +  24  d^^^^^^^  _  iq  ^3j4^8  _|_  ^g  ^4^,2^6^ 

11.  9a^-12  a%^  +  4  &«  +  24  aV  - 16  b^c"^  + 16  c«  -  30  aH 
+  2()bH-4.0cH-\-2^d\ 

12.  4  x^y"^  -  12  xHf  +  9  i«y  +  ^.x'^ifz  —  Qxh/z  +  x'^y'^z'^  - 
16  a;3z/2^8  +  24  xy^z^  -  8  cc^^^*  +  16  i/z^  +  4  x^'yz  -  6  ic^^/^^  + 
2  cc^T/^^  —  8  £c'^2/s*  +  x'^z'^. 

13.  1  +  a;  to  4  terms. 

14.  1  —  2  ic  to  4  terms. 

15.  4  +  2  a;  to  4  terms. 

16.  9  a^  +  12  ax  to  2  terms. 

17.  Find  x  so  that  a*  +  6a8  +  7a2  —  6<^4-a;  shall  be  a 
perfect  square. 

18.  Find  m  so  that  4  cc*  +  4  cc^  +  mx^  4-  4  cc  +  4  shall  be 
a  perfect  square. 

19.  Find  m  so  that  9  a*  +  12  a«  +  10  a^  +  w«^  +  1  shall 
be  a  perfect  square. 

20.  Show  that  the  square  root  of  2\_{in  +  ny  +  (m^  +  n*)'] 
is  2  (m^  +  71^  +  m?z). 


FACTORS.  99 


131.   The  square  roots  of  numbers  are  similarly  found. 
Kequii-ed  the  square  root  of  547.56. 

Eoot     =2    3.    4 

Power  =  5'47.56  contains  p  +  2fn  +  n^ 
f^  =  4  00.00 


2/i           =40  147.56         "         2/i%  +  V  /i  =  20 

2/,  +  %  =  43  1  29.00        = "  %=    3 

2/2           ==46  18.56  contains  2/2/^2  +  ^2^^  /2  =  23 

2/2  +  ^2  -  46.4  18.56        = "-  n^=    0.4 

Explanation.  1.  •.•  the  highest  order  of  the  power  is  lOO's,  the 
highest  order  of  the  root  is  lO's,  and  it  is  unnecessary  to  look  below 
lOO's  for  the  square  of  lO's. 

2.  Similarly,  it  is  unnecessary  to  look  below  I's  for  the  square  of 
I's,  below  lOOths  for  the  square  of  lOths,  etc. 

3.  The  greatest  square  in  the  lOO's  is  400,  which  is  the  square  of 
20,  which  may  be  called  /i  (read  "/-one  "),  the  first  found  part. 

4.  Subtracting,  147.56  contains  2fn  +  w^,  because /^  has  been  sub- 
tracted from/2  ^  2fn  -\-  n^,  where  /  stands  always  for  the  found  part 
and  n  for  the  next  order  of  the  root. 

5.  2fn  +  w2  is  approximately  the  product  of  2/  and  n,  and  hence, 
if  divided  by  2/,  the  quotient  is  approximately  n.     .-.  n  =  3. 

6.  .-.  2/  -f  n  =  43,  and  this,  multiplied  by  n,  equals  2/n  -|-  n^. 

7.  •••  /2  has  already  been  subtracted,  after  subtracting  2fn  -{-  n^ 
there  has  been  subtracted  p  -f  2/w  +  n^,  or  (/  +  n)2,  or  232. 

8.  Calling  23  the  second  found  part,  /2,  and  noticing  that 
f2=fi  +  m,  it  appears  that  23^,  or  f^^,  has  been  subtracted. 

9.  .-.  the  remainder  18.66  contains  2/2^2  -1-  ^2^. 

[10.  Dividing  by  2/2  for  the  reason  already  given,  712  =  0.4. 
[11.  .-.  2/2  +  ?i2  =  46.4,  and  18.56  =  2/2^2  +  n2^,  as  before. 
^12.    Similarly,  the  explanation  repeats  itself  after  each  subtraction. 

EXERCISES.    LI. 

Extract  the  square  roots  of  exs.  1-6. 
1.    958441.  2.   7779.24.  3.   32.6041. 

4.    24.1081.  5.    0.900601.  6.    0.055696. 


100  ELEMENTS  OF  ALGEBRA. 

132.   Cube  root  by  the  formula  f ^  +  3  f  ^n  +  3  fn^  +  n^. 
Eequired  the  cube  root  oi  S  a^  —  12  a^b  +  6  ab^  —  b^. 
Let  /  =  the  found  part  of  the  root  at  any  stage  of  the 

operation,  and 

n  =  the  next  term  to  be  found. 

Then  (/  +  ny  =/'  +  Sfhi  +  3fn^  +  n\  §  82 

The  work  may  be  arranged  as  follows : 

Root    =  2  a  —  6 

Power  =  8 a^ -12  a^b-\-Q  a¥-¥  contains 


P  = 

8a3                                                   /3+ 3/2^  4.  3/^2 +^3 

3/2 

3/n 

3/2  +  3/n 

-12  a26+6  a62-63  contains 

+n2 

+  n2 

3/2n+3/n2+n3 

12  a2 

-6ab 

12a2-6a6 

-12a26+6a62-63      = 

+  62 

+  62 

Explanation.     1.   The  cube  is  arranged  according  to  the  powers  of 
a  and  6  for  a  reason  similar  to  that  given  in  square  root. 

2.  •.■  8a^  =  the  cube  of  the  first  term,  the  first  term  is  2  a. 

3.  Subtracting  f^,  the  remainder,   —  12  a26  +  6  a62  —  6^,  contains 
3/2n  +  3/n2  +  n^ 

4.  Dividing  by  3/2  {i.e.,  12  a^),  n  is  found  to  be  —  6. 

5.  •.•/=  2  a,  and  n  =  -  6,  .-.  3/2  +  3/n  +  n2  =  12  cfi  -  6  a6  +  62. 

6.  Multiplying  by  w,  -  12  a26  +  6  a62  -  6^  must  equal  3/%  +  3/w2 
+  n^.     This  together  with/^  completes  the  cube  of  /  +  n. 

Check.     Let  a  =  6  =  1.     Then  is  =:  8  -  12  +  6  -  1  =  1. 

EXERCISES.    LII. 

Extract  the  cube  roots  of  exs.  1-6. 

1.  8a8-36a26+-54a&2_27^'^ 

2.  a^x''  -  12  a'bx''  +  48  ab^x''  -  64  b''x\ 

3.  1  -  6ic  +  2l£c2  -  44£c8  +.  63£c*  -  Ux^  +  27 (r^ 

4.  a^-2  a^b  +  I  a'^b^  -  f  0  a^""  +-  /^  ^'^^  -  /t  «^'  +  7k  ^'• 

5.  ^6  - 12  a%  +-  54  a'^^^^  _  1^2  a^Z*^  +- 108  a^J*  -  48  ab^  +  8  b\ 

6.  iB8+3a;22^_e^2_^3^2/2_12cc?/+-12ic+2/^-6/+l27/-8. 


FACTORS. 


101 


133.    The  cube  roots  of  numbers  are  found  by  the  same 
general  method. 

W      Eequired  the  cube  root  of  139,798,359. 
Root     =519 

Power  =  139,798,359  cont's/3+3/2n+3/n2  +  ?i8 
P  =  125,000,000 


3/2 

L 

3/n 

+  n2 

3/2  +  3/n 

+  n2 

14,798,359  contains  3/2n  +  SM  +  n^ 
/i  =  500 

750,000 

15,100 

765,100 

7,651,000  =  3/2n  +  3/w2  +  n^ 

780,300 

13,851 

794,151 

7,147,359  contains  3/2n  +  3/n2  +  n^ 

h  =  510 
7,147,359  =  3/2n  +  3/n2  +  n^ 

na  =9 

Explanation.  1.  •.•  the  highest  order  of  the  power  is  hundred- 
millions,  the  highest  order  of  the  root  is  lOO's  (why  ?),  and  it  is  unnec- 
essary to  look  below  millions  for  the  cube  of  100 's.     (Why  ?) 

2.  Similarly,  it  is  unnecessary  to  look  below  lOOO's  for  the  cube  of 
lO's,  below  I's  for  the  cube  of  I's,  etc. 

3.  The  greatest  cube  in  the  hundred-millions  is  125,000,000,  the 
cube  of  500.     .-.  500  may  be  called  /. 

4.  Subtracting,  14,798,359  contains  3/2n  +  3/n2  -|-  n^.     (Why  ?) 

5.  This  is  approximately  the  product  of  3/2  and  w,  and  hence  if 
ivided  by  3/2  the  quotient  is  approximately  n.     .-.  w  =  10. 

6.  .-.  3/n  +  n2  =  15,100,  and  3/2  +  Sfn  +  n2  =  765,100,  and  this, 
multiplied  by  n,  equals  3/2n  -|-  3/n2  +  n^. 

7.  •••  P  has  already  been  subtracted,  after  subtracting  3/2w  +  3/n2 
I    +n^  there  has  been  subtracted  (/  -}-  n)%  or  510^. 

8.  Calling  510  the  second  found  part,  /a,  it  appears  that  fz^  has 
'A   been  subtracted.     .-.  the  remainder  contains  3/2n  +  3/n2  +  n^. 

■ft  9.   The  explanation  now  repeats  itself  as  in  square  root. 

II 


102  ELEMENTS  OF  ALGEBRA. 

EXERCISES.    T.TTT 

Extract  the  cube  roots  of  exs.  1-4. 

I.  (a)  10,077,696.     (b)  31,855,013.      (c)  125.751501. 

2.  (a)  367,061.696.         (b)  997.002999. 

3.  (a)  551.     (b)  975.     Each  to  0.001. 

4.  (a)  2.         (b)  5.         Each  to  0.0001. 

BEVTEW   EXERCISES.    UV. 

Extract  the  cube  roots  of  exs.  1-3. 

1.  1  —  a;  to  5  terms. 

2.  64  —  48  X  +  9  arHo  3  terms. 

3.  a»  +  9  a^b  +  36  a?b''  +  84  a«ft»  +  126  a^h^  +  126  a*** 

+  84  a*b^  +  36  a%''  +  9  a*«  +  b\ 

4.  Factor  x*  +  a;^  —  4ic*  —  4. 

5.  Show  that  xyz  (x^  +  y*-\-z^  —  {fz^  +  s*a;«  +  x^f)  = 
(x^  —  yz)  {f  —  zx)  (z^  —  xy). 

6.  Divide  the  product  ol  x^  -\-x  —  2  and  x*  +  a:  —  12  by 
the  sum  oi  2  x^ -\- 6x  -^  1  and  2  —  x  (10  +  x). 

7.  Find  the  square  root  of 

(x  +  3)  (X  +  4)  (x  +  5)  (X  +  6)  +  1 . 

8.  Solve  the  equation 

7  -  2  J6  -  3[5  -  2(4  -  3  +  2x)];  =  1. 

9.  Find  the  square  root  of 

(2  a-  by  -2(2a^  -5ab  +  2b^  +  (a  -2by. 
10.   Find  the  three  roots  of  the  equation  x^  —  x^  -\- 1  =  x. 

II.  Also  of  the  equation  a;»  +  9  x^  +  8  a;  -  60  =  0. 

12.    If  a  =  —  3,  5  =  0,  c  =  1,  <^  =  —  2,  find  the  numerical 
value  of  a- 2\b  + Sic -2a -(a- b)]-\- 2  a -(b-\-3c)\. 


CHAPTER   Vn. 

fflGHEST   COMMOX   FACTOR    AXD  LOWEST 
COMMON  MULTIPLE. 

L    m^SST  COMMON  FACTOR. 

134.  The  integral  algebraic  factor  of  highest  degr^ 
cammon  to  two  or  more  integral  algebraic  expressions  is 
called  their  IrighrBt  iwiibim  ^Mrtor. 

S.g.,  o^  is  the  higiwst  common  tictor  ol     c\riauid2an^, 

a-6     "  "  "     {a-Hf  "    rt*-6». 

Consider,  abo,  2(«»-6»)aiid4(6«-i^. 

Here  2  («»  -  6»)  =  2  (a  -  6)(a2  +  oft  +  ft*),  or  -  2<&-a)(tf>+afr+6^r 
and        4<^-ai*)  =  4(6-a)(6  +  o),  •  -4(a-6)(a+6). 

-6or6  —  aisa  commnn  fatAar,  and  there  hei^g  no 

edraic  fMCtor,  either  is  cdkd  tike  h^^iert  eoHBon 

uor.     Tiicre  La  a  commm  nnmerieal  fKtor,  2,  hot  audi  fattaa  hsie 

hing  to  do  with  the  a^efaiaic  dir^iMli^  of  tiie  ezpraaaioDs,  and 

nee  may  be  Defected. 

In  the  last  example,  it  is  iK>t  osnal  to  state  both  aBSWOS,  a  —  b  and 

-  a,  because  a  —b=  — 1(6  —  a);  that  is,  the  two  are  the  same 

pnr  tor  a  nnmeiical  factor,  and  numencal  £Mt(ffs  are  not  eon- 


135.   The  arithmetical  greatest  common  diTisor  must  not 
confounded  with  the  algebraic  highest  common  factor, 
ough  these  are  often  called  by  the  same  name.     The 
common  factor  has  reference  only  to  the  degree  of 
erpression. 

103 


-  conf 

iKhest 
IKexp 


104  ELEMENTS  OF  ALGEBRA. 

E.g. ,  consider  the  highest  common  factor  otx^  —  Sx  +  2  and  x^—x-2. 

Here        a;2  -  3x  +  2  =  (x  -  2)  (x  -  1),  or      (2  -  x)  (1  -  x), 
and  x2-x-2  =  (x-2)(x  +  l)    "  -  (2  -  x)  (x  +  1) ; 

hence,  the  highest  common  factor  is  x  —  2,  or  2  —  x.  Now  if  x  =  5, 
the  expressions  become  12  and  18,  and  the  highest  common  factor 
becomes  3,  or  —  3,  although  6  is  the  greatest  common  divisor  of  12 
and  18. 

The  highest  common  factor  is  occasionally  used  in  reduc- 
ing fractions  to  their  lowest  terms. 

136.  Factoring  method.  The  highest  common  factor  of 
expressions  which  are  easily  factored  is  usually  found  by 
simple  inspection. 

E.g. ,  to  find  the  highest  common  factor  of  x2  —  3 x  +  2,  x^  —  x^  -  2 x, 
and  i  x2  +  I X  —  3,  we  have  : 

1.  x2  -  3x  +  2  =  (X  -  2)  (X  -  1). 

2.  x3-x2-2x  =  x(x-2)(x  + 1). 

3.  ix2  +  |x-3  =i(x-2)(x  +  3). 

4.  .-.  the  highest  common  factor  is  x  —  2,  or  2  —  x. 

EXERCISES.    LV. 

Pind  the  highest  common  factor  of  each  of  the  following 
sets  of  expressions : 

2.  15  mnx^,    17  mx^yz,    f  abcx^^z. 

3.  l()xhjz,    Ibaxhjz^,    20  amxz'\ 

4.  x^  —  2/^,    if  —  x^,    x'^  —  %xy  +  1  y^. 

5.  x^  —  y^,    2/^  — a^^    a^^  — i^y  —  iy^- 

6.  x''-4.,    x'^-x-Q>,    2-5x-Sx''. 

7.  2x^  —  xy  —  y^,    4:  x^ -\- 10  xy -\- 4:  y\ 

8.  6a^  +  19ab-7b%    2  a^ -\- ab  -  21  b\ 

9.  4a2(et8-68)^    ^ab''{3a''-^ab-{-2b'). 


FACTORS   AND   MULTIPLES. 


105 


137.  If  the  factors  of  one  of  several  algebraic  expressions 
are  known,  but  those  of  the  others  not,  it  is  easy  to  ascer- 
tain, by  division  or  by  the  Remainder  Theorem,  if  the  known 
factors  of  the  one  are  factors  of  the  other. 

E.g.,  to  find  the  highest  common  factor  of  1  —  cc^  and  llSx'^  —  4x' 
+  2X-11L 

Here  1  -  x2  =  (1  -  x)  (1  +  x),  or.-  (x  -  1)  (x  +  1). 

But  X  —  1  is  a  factor  of  113 x'^  —  4 x^  +  2  x  —  111,  by  the  Remainder 
Theorem  (§  103),  while  x  +  1  is  not.  .-.  x  —  1  is  the  highest  common 
factor. 


I 


EXERCISES.    LVI. 


rind  the  highest  common  factor  of  each  of  the  following 
sets  of  expressions : 


1. 

x'  -  //^  x^  -  /. 

2. 

x^-4.,  a;^- 4x2 -16. 

3. 

^2-4,  x^  +  7x2  +  100. 

4. 

cc^  +  1,  x^  +  ax^  +  ax  +  1. 

5.  x^  -  3  X  4-  2,  x2  -  9  X  +  14. 

6.  x2-9x  + 14,  2x^-5x2-441. 

8.  x2  -  4,  5  X*  +  2  x^  -  23  x2  -  8  X  +  12. 

9.  2x2 -5x7/ +  32/2,  6x3 -23x^2/ +  25x7/2 -6/. 
(10.  ^8  _  53^  yi  _  ^2^  117  ^3  _  117  ^2^  _  231  ab  +  231  h\ 
111.  x^-l,  x2-l,  293xS-200x'^  +  7x8-50x2-25x-25. 
[12.  1-x^  x^-1,  x^-l^Zx-Zx",  247x2-240x-7. 
[13.  x^  -  32,  16  -  X*,  x2  -  9x  +  14,  X*  -  4x2  +  6x  -  12, 
[14.  x«  +  1,  x2  +  2x  +  1,  x^  +  1,  324xs  ^  2^r^r^^  j^  100x» 

+  204x2-27. 


106  ELEMENTS   OF  ALGEBRA. 

138.  Euclidean  method.  In  case  the  highest  common  fac- 
tor is  not  readily  found  by  inspection  of  factors,  a  longer 
method,  analogous  to  one  suggested  by  Euclid  (b.c.  300) 
for  finding  the  greatest  common  divisor,  may  be  employed. 

139.  This  method  depends  upon  two  theorems : 

1.  A  factor  of  an  algebraic  expression  is  a  factor  of  any 
multiple  of  that  expression. 

Proof.  1.  Let  a,  6,  p,  q  be  algebraic  expressions,  p  and  q  being 
the  factors  of  b. 

2.  Then  6  =  pq. 

3.  .-.  ab  =  apq.  (Why  ?) 

4.  I.e.,  if  p  is  a  factor  of  b,  it  is  a  factor  of  any  multiple  of  &,  as  ab. 

A  similar  proposition  is  readily  seen  to  be  true  for  num- 
bers. E.g.,  5  is  a  factor  of  35 ;  and  since  multiplying  35 
by  any  integral  number  does  not  take  out  this  5,  therefore, 
5  is  a  factor  of  any  multiple  of  35. 

2.  A  factor  of  each  of  two  algebraic  expressions  is  a  factor 
of  the  sum  and  of  the  difference  of  any  multiples  of  those 
expressions. 

Proof.     1.   Let  b  =  pq    and    6'  =  pq'. 

2.  Then  ab  =  apq    "    a'6'  =  a'pq'.  (Why  ?) 

3.  .-.  ab  ±  a'b'  =  apq  ±  a'pq'  =p{aq  ±  a'q').  (Why  ?) 

4.  I.e.,  if  p  is  a  factor  of  b  and  b\  as  in  step  1,  then  it  is  also  a 
factor  of  the  sum  and  of  the  difference  of  any  multiples  of  b  and  6', 
as  ab  and  a'b'. 

A  similar  proposition  is  true  for  numbers.  E.g.,  5  is  a 
factor  of  60  and  of  35,  and  also  of  the  sum  and  of  the  dif- 
ference of  any  multiples  of  these  numbers. 

140.  The  Euclidean  method  will  best  be  understood  by 
considering  an  example. 


FACTORS  AND  MULTIPLES.  107 

Required  the  highest  coninion  factor  of 

x^  —  x^  -{-2x^  —  x  +  1  and  x^  +  x^  -^  2x^  +  x -i-1. 

x4-x3+2x2-ic+l|a:44-    x^+2x'^+    x  +  l[l 
x4-    x^+2x^-    x  +  1 
2a;l2x3  +2x 

x2  +I|x4-x3+2x2-ic+llx2-a;+l 


x* 

+ 

X2 

-X3  + 
-X3 

X2- 

-x+1 

-X 

X2 
X^ 

+  1 

+  1 

Explanation.  1.  The  h.c.f.  of  the  two  expressions  is  also  a  factor 
of  2x3+2x,  by  th.  2  (§  139). 

2.  It  cannot  contain  2  x,  because  that  is  not  common  to  the  two 
expressions. 

3.  .-.  2  X  may  be  rejected,  and  the  h.c.f.  must  be  a  factor  of  x2  +  L 

4.  x2  -f  1  is  a  factor  of  x*  —  x^  -f  2  x2  —  x  +  1,  by  trial. 

5.  "        "  "        2x3 +  2x. 

6.  .-.      "        "  "        x* +  x3  +  2x2  +  x  + 1.  (Why?) 

7.  .-.      "      is  the  h.c.f.  (Why?) 

141.  In  order  to  avoid  numerical  fractions  in  the  divi- 
sions, it  is  frequently  necessary  to  introduce  numerical 
factors.  These  evidently  do  not  affect  the  degree  of  the 
highest  common  factor. 

E.g.,  to  find  the  highest  common  factor  of  4x3  —  12x2  _|_  n^  —  3 
and  6x3-  13x2  +  9x  -2. 

6x3-13x2+  9x-2 

2 

4x3-12x2  +  llx-3|l2x3-26x2  +  18x-4[3 
12x3--36x2  +  33x-9 
5|l0x2-15x  +  5 

~   2x2-  3x+l|4x3-12x2+llx-3|2x-3 
'      "  4x3-  6x2+  2x 

-  6x2+  9a;_3 

-  6x2+  9a;-3 

Here  the  introduction  of  the  factor  2  and  the  suppression  of  5  evi- 
dently do  not  affect  the  degree  of  the  highest  common  factor. 


108  ELEMENTS  OF  ALGEBRA. 

142.  In  practice,  detached  coefficients  should  be  used 
whenever  the  problem  warrants. 

E.g.,  to  find  the  highest  common  factor  of 
3  xSy  +  3  x^y  +  2  x^y  -  x'^y  -  xy  and  2x'^  +  9x^  +  9x'^  +  1  x. 

Here  x  is  evidently  a  factor  of  the  highest  common  factor.  It  may 
therefore  be  suppressed  and  introduced  later,  thus  shortening  the 
work. 

But  ?/  is  a  factor  of  the  first  only,  and  hence  may  be  rejected 
entirely. 

The  problem  then  reduces  to  finding  the  highest  common  factor  of 
3x*  +  3x3  +  2x2-x-  1  and2x3  +  9x2  +  9x  +  7. 


3+3  + 

2- 

1  - 

1 
2 
2|3 

2  +  9  +  9  +  7|6+  6  + 

4- 

2  - 

6  +  27  + 

27  + 

21 

-21  - 

23- 

23- 

2 

2 

42  + 

46  + 

46  + 

4[21 

42  +  189  +  189  +  147 

-  143l  - 

143- 

143- 

143 

1  + 

1  + 

l|2  +  9  +  9  +  7|2  +  7 
2  +  2  +  2 

7  +  7  +  7 

.-.  x(x2  +  x  +  1)  istheh.c.f. 

7+7+7  ' 

143.    The  work  can  often  be  abridged  by  noticing  the  dif- 
ference between  the  two  polynomials. 

E.g.,  m  the  case  of  x*  -  2x3  +  Sx^  -  8x  +  6  and  x*  -  4x3  +  3x2  _ 
6  X  +  6.     Here  we  have  : 


1 
1 

-2  +  3- 
-4  +  3- 

-8  +  6 
-6  +  6 

2)2 

-2 

1       - 1 

X2  -  1  =  (X  +  1)  (X  -  1). 

By  the  Remainder  Theorem  x  -  1  is  a  factor  of  each  expression,  and 
X  +  1  is  not ;  .-.  x  —  1  is  the  highest  common  factor  of  the  expressions. 


FACTORS  AND   MULTIPLES. 


109 


144.  The  highest  common  factor  of  three  expressions 
cannot  be  of  higher  degree  than  that  of  any  two ;  hence, 
the  highest  common  factor  of  this  highest  common  factor 
and  of  the  third  expression  is  the  highest  common  factor  of 
all  three.     Similarly,  for  any  number  of  expressions. 


EXERCISES.    LVII. 

Find  the  highest  common  factor  of  each  of  the  following 
sets  of  expressions : 

1.  ic»  —  2  ic  +  4,  x^  -\-x^  +  4:X. 

2.  2x^  +  2x-4.,  x^-3x-{-2. 

3.  x^  +  4.,  x^-2x^'{-x^  +  2x-2. 

4.  x^-4:0x  +  63,  x^-7x^-h63x-Sl. 


5.    x^  +  y^ 


2/^  x'  +  xV  +  xy  +  y\ 


6.  x«(6a:  +  l)-£c,  4£c»-2cc(3£c  +  2)  +  3. 

7.  £c4-15£c2  4-28x-12,  2ic«-15cc  +  14. 
Ks.  Ix^  -\^x'-1x^\^,2x^-x''-2x^\. 
IH  9.  x^-^x-  117,  a;*  -  13ic8  -  a;2  +  I4aj  -  13. 
I      10.  63  a*  -  17  a^  +  17  a  -  3,  98  a"  +  34  a^  +  18. 

11.  a;2_^4a^-21,  cc2  +  20£c  +  91,  2x'' ^ix^ -'l^  x. 

12.  '^x^-l^x^^lx''-2x,  6£c5-llic*  +  8iK»-2ic2. 

3.  9a2-4^>2  +  4^>c-c2,  2^  ^  c" -\-Z  ah -3hc -3ac. 

4.  {a  -  b)  {a^  -  c2)  -(a-c)  {a^  -  b^),  a^  -  b',  ab  -  b^ 
—  ac  -\-  be. 

5.  cc3-10(a;2  +  3)+ 31a!,    ^^^(x  -  11)  +  2(19  :b  -  20), 
a;3_9cr2  +  26£c-24. 

6.  a*52  +  4  a^b^  +  3  a%''  -4.ab^-4.  b\  a^'b  +  3  a%^  -  a'^h^ 

7.  3a^-7  ab  +  2b^  +  bac-6bc  +  2c%  12  a^  -  19  ab 
+  5b''  +  llac-llbc  +  2c^. 


110  ELEMENTS   OF  ALGEBKA. 

II.     LOWEST   COMMON  MULTIPLE. 

145.  The  integral  algebraic  multiple  of  lowest  degree 
common  to  two  or  more  algebraic  expressions  is  called 
their  lowest  common  multiple. 

E.g.^  a^b^cd  is  the  lowest  common  multiple  of  a^bc  and  ab^^d. 
Similarly,    ±  (a  +  b)^  (a  —  b)   is  the  lowest  common  multiple  of 
a2  -b%b  -  a,  and  (a  +  6)2.     For 

1.  a2  -  62  =  (a  +  6)  (a  -  6). 

2.  6  -  a  =  -  (a  -  6). 

3.  (a  +  6)2  =  (a  +  6)  (a  +  6). 

4.  .-.  either  (a  +  6)2  (a  —  6)  or  (a  +  6)2  (6  —  a)  contains  the  given 
expressions  and  is  the  common  multiple  of  lowest  degree. 

The  lowest  common  multiple  of  algebra  must  not  be  con- 
sidered the  same  as  the  least  common  multiple  when 
numerical  values  are  assigned,  ^.ff.,  the  lowest  common 
multiple  of  a  -\-  b  and  a  —  b  is  (a  -\- b)  (a  —  b)  ;  but  it  a  =  6 
and  b  =  4:,  the  least  common  multiple  of  6  +  4  and  6  —  4  is 
simply  6  +  4. 

146.  So  far  as  the  algebraic  multiple  is  concerned,  numer- 
ical factors  are  not  usually  considered. 

E.g.,  aWc  is  the  lowest  common  multiple  of  2  ab^c,  icfib,  and  15 a6. 

The  lowest  common  multiple  is  used  in  reducing  fractions 
to  fractions  having  a  lowest  common  denominator. 

147.  Factoring  method.  The  lowest  common  multiple  is 
usually  found  by  the  inspection  of  factors. 

E.g.,  to  find  the  lowest  common  multiple  of  x2  _  12  x  +  27,  x2  +  x 
-  12,  and  15  -  2  X  -  x2. 

.  1.  x2  _  12  X  +  27  =      (X  -  3)  (X  -  9). 

2.  x2  +  X  -  12  =       (X  -  3)  (X  +  4). 

3.  15-2x-x2=  -(x-3)(x  +  5). 

4.  .-.  -t  (x  —  3)(x  +  4)(x  +  5)(x  —  9)  is  the  lowest  common  multiple. 
In  practice,  the  result  should  be  left  in  the  factored  form. 


FACTORS  AND   MULTIPLES.  Ill 

EXERCISES.    LVIII. 

Find  the  lowest  common  multiple  of  each  of  the  follow- 
ing sets  of  expressions : 

1.  —lOa^xyz,  hxSjz^j  \a^xy^z. 

2.  x^  +  2/^  ^  +  2/j  xy  —  x^  —  y^. 

3.  a^  4.  ^2  _  2  ab,  52  _  ^2^  a-  h. 

4.  27  -  12a^  +  a;2,  ic2  +  2x  -  15. 

5.  ic*  +  4,  2  -  cB^^  a;2  +  2,  ic  -  V2. 

6.  a^2  ^  ic  -  12,  -  36  +  13  X  -  ic2,  £c2  -  16. 

7.  ic^  +  2/^  +  3  £c?/  (cc  +  ?/),  cc^  +  2/^  ^  +  2/- 

8.  2  ic?/  —  ic^  —  ?/^,  2xy  -\-  x'^  -\-  y'^,  x^  —  y^,  x  ■\-  y. 

148.  Highest  common  factor  method.  Since  the  highest 
common  factor  contains  all  of  the  factors  common  to  two 
expressions,  it  may  be  suppressed  from  either  of  them  and 
the  quotient  multiplied  by  the  other  to  obtain  the  l.c.m. 

Proof.     1.  Let  x  —  af, 

y  =  ¥, 

in  which  /  is  the  highest  common  factor  of  x  and  y. 

2.  Then  the  lowest  common  multiple  is  evidently  abf\ 
i.e.,  it  is  y  multiplied  by  a. 

E.g.,  to  find  the  lowest  common  multiple  of  2^3  +  Sx^  —  3x  —  27 
and  2  x3  +  12  x2  +  X  -  45. 

2x3  +  12x2+    x-45 
2x3+    8x2-3x-27 


2 1 4  x2  +  4  X  - 

-18 

2x2 +  2x- 

-    9|2x3  +  8x2-3x-27|x  +  3 

2  x3  +  2  x2  -  9  X 

6x2  +  6x-27 

6x2  +  6x-27 

:-  (2  x3  +  12  x2  +  X  —  45)  (x  +  3)  is  the  lowest  common  multiple. 


112  ELEMENTS  OF  ALGEBRA. 

EXERCISES.  LIX. 

Find  the  lowest  common  multiple  of  the  sets  of  expres- 
sions in  exs.  1-15. 

1.  a;i2  +  a;^  x^''  +  x\ 

2.  3  a«  -  11  ^2  +  4,  6  a2  -  a  -  2. 

3.  x'  +  Sx'  +  x  +  S,  x^-8x  +  3. 

4.  6a;2  +  13x  +  6,  10cc2-3  +  13a;. 

5.  x^  +  2ax  +  a%  x^  +  ab -^  (a  +  b)x. 

6.  Gx^  +  llx^'-dx  +  l,  2a;2  +  3x-2. 

7.  x^-x^  +  x^  —  x-4.,x^-x^  +  2x-8. 

8.  x^  +  l  +  3{x^  +  x),  x^  +  l  -^4:{x^  +  x)-{-6x^ 

9.  3x^-15ax^  +  a^x-5a^,  6£c*  -  25aV  -  9  a*. 

10.  x^  +  20x  +  91,35-2x-x'',x^  +  6x^-6x''  +  6x-7. 

11.  2a;^-2x3-a;2-4a;-7,  2ic*  +  6ic«-17£cH8a;-35. 

12.  x^  +  x^  +  x  +  1,    2 x^  -  3 x^  -\-  4:x^  +  2 x^  -  3  X  +  4:. 

13.  a;^  +  a;«-a;*-^6£c2_5^_7^  ic^  -  x«  -  ic^  +  ic*  -  6 a;» 
—  X  +  7. 

14.  cc"^  +  2a;«  -  3  £c^  +  ic^  +  2 a;  -  3,  x^  +  4 a;^  -  7 cc^  +  a;^ 
+  4(K-7,  x^  +  1, 

15.  4^2(3  ^^2) -(27  a +  18),  12a3- a(8a  +  27)  +  18, 
6(3a-2)  +  27a«-8. 

16.  Find  all  of  the  algebraic  expressions  whose  lowest 
common  multiple  is  ic^  —  4  xy\ 

17.  Prove  that  the  product  of  the  lowest  common  mul- 
tiple and  the  highest  common  factor  of  two  expressions  is 
the  same  as  the  product  of  the  two  expressions. 

18.  Investigate  ex.  17  for  the  case  of  three  expressions. 

19.  Find  the  lowest  common  multiple  of  a^  —  1  and 
a^  —  4  a  +  3.  Can  the  result  be  checked  by  letting  a  =  5, 
7,  or  any  odd  number  above  3  ?     Explain. 


FACTORS  AND   MULTIPLES.  113 

REVIET77   EXERCISES.    LX. 

1.    Factor  x(x  —  1)  —  a(a  —  1). 

12.    Solve  the  equation  4  ic^  +  1  =  4  cc. 
3.    Solve  the  equation  6x^  +  llx  —  7  ~0. 
4.    Extract  the  square  root  of  cc^  +  1  to  3  terms. 
5.    Give  a  complete  description  of  this  expression  as  a 
function  of  x  and  y:    x^  -{-  3  xSj  +  4  xhj'^  +  3  xy^  +  ?/*. 

6.  Show  that  the  difference  of  the  squares  of  any  two 
consecutive  numbers  is  equal  to  the  sum  of  the  numbers. 

7.  Find  the  lowest  common  multiple  of  2x^  -\-  x'^  -\-  ^x^ 
-{-4.x^  +  2x  +  S  and  6x^  -  5x^  +  12x^  -  Sx^  +  5x  -  6. 

8.  Find  the  lowest  common  multiple  otx^  —  x^  —  2x  —  l, 
2x''-x'^-2x'-2x-l,  and  Zx^  _  4a;8  +  6a;2  -  7  a;  -  8. 

9.  Find  the  highest  common  factor  of  x*  +  2  ic^  —  5  a;^  -}- 
15a; +  12,  a;4  +  5a;3  +  5cc2^8ic  +  16,  and  ic^  +  6 a;8  +  10 a;^ 
+  4a;-16. 

I^B-10.    In  finding  the  highest  common  factor  of  two  alge- 
■flfi'aic  expressions,  by  what  right  may  a  factor  be  suppressed 

I  in  one  if  it  is  not  a  factor  of  the  other  ? 
I^B  11.    The  highest  common  factor  of  two  expressions  is  4  x"^ 
^^  a?-,  and  their  lowest  common  multiple  is  4a;'^  —  ha^x^  +  a*. 

j  One  of  the  expressions  is  4  a;^  +  4  ax'^  —  a^x  —  a^.     Find  the 
other. 

12.  Assign  such  values  to  a  and  h  that  the  arithmetical 
least  common  multiple  of  a^  —  b^  and  a^  -\-  b^  -\-  2  ah  (a -{-  b) 
shall  not  be  the   value  of  the  algebraic  lowest  common 

|ig|ultiple. 

I^rl3.  Prove  that  the  difference  between  the  cubes  of  the 
sum  and  difference  of  any  two  numbers  is  divisible  by  the 
sum  of  the  square  of  the  smaller  number,  and  three  times 
the  square  of  the  lai-ger. 


■ 


CHAPTER   VIII. 
FRACTIONS. 

149.   The  symbol  -?  in  which  b  is  not  zero,  is  defined  to 

mean  the  division  of  a  by  b,  and  is  called  an  algebraic 
fraction. 

Hence,  the  algebraic  fraction  -  represents  a  quantity 

which,  when  multiplied  by  ft,  produces  a. 

The  terms  of  the  fraction  -  are  a  and  b,  a  being  called  the 

numerator  and  b  the  denominator,  and  either  or  both  may  be 
fractional,  negative,  etc. 

The  case  in  which  h  equals  zero  is  discussed  later. 

There  are  two  definitions  of  a  fraction  usually  given  in 
arithmetic :  (1)  The  fraction  y  is  a  of  the  b  equal  parts  of 
unity ;   (2)  The  fraction  -  is  one  bth.  of  a. 

Neither  of  these  arithmetical   definitions  includes,    for  example, 
2      2      3 

,  -,  -y=,  etc.,  for  "2  of  the  —  3  equal  parts  of  unity  "  means 

-3    f     v2 

nothing,  and  "one  V2th  of  3"  is  equally  meaningless.     Hence  the 

broader  algebraic  definition. 

In  the  first  arithmetical  definition  above  given,  6  names  the  part 
and  hence  is  called  the  denominator  (Latin,  namer),  and  a  numbers 
the  parts  and  hence  is  the  numerator  (Latin,  numberer).  Hence  the 
origin  of  these  terms. 

The  fraction  -  is,  therefore,  read  "a  divided  by  6,"  although  the 

reading  "a  over  6"  is  generally  used  in  various  languages,  and  is 
sanctioned  by  most  teachers  on  the  ground  of  brevity. 

114 


FRACTIONS.  115 


I.     REDUCTION   OF  FRACTIONS. 

150.  Theorem  of  reduction.  The  same  factor  may  be  intro- 
duced into  or  cancelled  from  both  numerator  and  denomina- 
tor of  a  fraction  without  altering  the  value  of  the  fraction. 

Given         the  fraction  -■>  and  m  any  factor. 

To  prove    that  -  =  — -j  that  is,  that  the  factor  m  may 
0      mo  a 

be  introduced  into  both  terms  of  -  or  cancelled 

from  both  terms  of  — r  • 
mo 

Proof.    1.  b--  =  a.  Def.  of  frac. 

b 

2.  .*.  mb--  =  ma.  Ax.  6 

b 

3..:  ^  =  ^.  .  Ax.  7 

b       mb 

An  algebraic  fraction  is  said  to  be  simplified  when  all 
common  algebraic  factors,  and  hence  the  highest  common 
factor,  of  both  numerator  and  denominator  have  been  sup- 
pressed, and  there  is  no  fraction  or  common  numerical 
factor  in  either. 

(j2  I  2  a6  4-  &2 

E.g.,  the  fraction  z — — is  simplified  when  reduced  to  the 

a^  +  0^ 

form  — ; —  by  cancellme  the  factor  a  +  b. 

J  a2  -  a6  +  &2    -^  & 

I^BBut  the  fractions  and  —  are  not  simplified. 

M  "  The  student  should  notice  that  the  theorem  does  not 
allow  the  cancellation  of  any  terms  of  the  numerator  and 
denominator.     No  factor  can  be  cancelled  unless  it  is  con- 


116  ELEMENTS   OF  ALGEBRA. 

Usually  the  factors  common  to  the  two  terms  of  the 
fraction  can  be  found  by  inspection  and  cancelled ;  other- 
wise the  highest  common  factor  of  both  terms  is  found  and 
then  cancelled. 

—  a^'^cd^ 
Examples.     1.  Simplify  the  fraction  —  • 

1.  Cancelling  a^^  52^  ^^  ^nd  d^,  the  fraction  reduces  to 

h 

2.  And  since  there  are  no  other  common  factors,  and  the  terms  are 
integral,  the  fraction  is  simplified. 

Check.     Let  a  =  3,  6  =  d  =  2,  c  =  1.     Then  "  27  •  4  •  1  •  16       -  6 


9.8.1.8  2 


2.  Simplify 


1.   This  evidently  equals 


{a  +  hY 


{a  +  6)  (a  -  h) 

a-\-h 


2.  Cancelling  a  +  6,  this  reduces  to 

a  —  0 

3.  And  since  there  are  no  other  common  factors,  and  the  terms  are 
integral,  the  fraction  is  simplified. 

Check.  Let  a  =  2,  &  =  1.  Then  f  =  f .  (If  a  and  h  are  given  the 
same  values,  the  denominator  becomes  zero,  a  case  excluded,  for  the 
present,  by  the  definition  of  fraction.) 

3.  Simplify  3^^ +  26. -77 


3x^-l{)x  +  l 


1.  A  factor  of  each  term  of  the  fraction  is  a  factor  of  their  differ- 
ence, 36X-84  (§  139,  2). 

2.  Hence  of  3  x  —  7,  because  the  terms  of  the  fractions  do  not  con- 
tain 12. 

3.  Hence,  if  there  is  a  common  factor,  it  is  3  x  —  7,  because  this  is 
irreducible. 

4.  By  substituting  arbitrary  values  this  is  seen  to  be  a  probable 


factor,  and  the  fraction  reduces  by  division  to 


x+  11 
x-1  ' 


Check.     Let  x  =  2.     (Why  not  1  ?)     Then  -^  =  — . 


FRACTIONS.  117 


4.  Simplify    o    <i  ,   A    2   I   rz nr ' 


1.  Here  the  simple  factors  are  not  as  easily  determined  as  the 
highest  common  factor,  x^  -\-  x  +  2. 

2  X  -\-  7 

2.  Cancelling  this,  the  fraction  reduces  to 

3x  +  l 

3.  .-.  the  fraction  is,  by  definition,  simplified. 
Check.     Let  x  =  1.     Then  ff  =  f . 

If  the  student  has  not  studied  Appendix  III,  ex.  5  may 
be  omitted. 


I 


.   Simplify  -a'{i-o)-!'Hc~a)-e^a-i) 
^     ^  (a  —  b)(b  —  c)  (c  —  a) 


.  By  the  Remainder  Theorem  (§  104)  a  —  &  is  a  factor  of  both  terms 
of  the  fraction.  (We  try  a  —  6  because  if  there  is  any  common  factor 
it  must  be  a  —  6,  6  —  c,  or  c  —  a.) 

2.  Hence,  because  both  terms  are  cyclic,  b  —  c  and  c  —  a  are  factors. 

3.  And  since  the  numerator  is  of  the  4th  degree,  the  other  factor  is 
a  linear  cyclic  factor.     Hence,  it  is  n(«  +  6  +  c). 

4.  Hence,  the  numerator  is n {a  +  b  +  c){a  —  b){h  —  c){c  —  a).  But 
by  substituting  the  values  a  =  2,  6  =  1,  c  =  0,  n  is  seen  to  be  1. 

5.  Hence,  the  fraction  equals  a  +  6  +  c. 

Check.     Let  a  —  S,  b  =  2,  c  =  l  (values  different  from  those  used 

—  12 
for  finding  n).     Then =  6. 

151.  General  directions  for  simplifying  fractions.  The  pre- 
ceding fractions  were  simplified  in  different  ways.  While 
there  is  no  general  method  of  attack,  and  the  student  must 
use  his  judgment  as  to  the  best  plan  to  pursue,  the  follow- 
ing directions  are  of  value : 

1.  Cancel  monomial  factors  first,  as  in  ex.  1. 

2.  Then  see  if  common  polynomial  factors  can  he  readily 
discovered.  Make  free  use  of  the  RcTnainder  Theorem.  Com- 
pare ex.  2. 


118  ELEMENTS  OF   ALGEBRA. 

3.  If  common  factors  are  not  readily  discovered,  see  if  the 
difference  between  the  numerator  and  denominator  can  be 
easily  factored.  If  so,  try  these  factors,  using  arbitrary- 
values  or  the  Remainder  Theorem,  as  in  ex.  3. 

4.  Never  perforvi  a  multiplication  until  compelled  to. 
Factor  whenever  possible.  If  the  terms  are  cyclic  and  you 
have  studied  Appendix  III,  apply  your  knowledge  of  sym- 
metry and  homogeneity,  as  in  ex.  5. 

5.  Let  the  Triethod  by  finding  the  highest  common  factor 
be  the  final  resort.  For  one  who  is  skillful  in  factoring, 
this  tedious  method  ought  rarely  to  be  necessary.  In  ex.  5 
students  will  probably  use  the  Remainder  Theorem  instead 
of  the  method  suggested. 

6.  Always  check  the  final  result  by  substituting  arbitrary 
values  or  by  some  other  simple  device. 

EXERCISES.    LXI. 

Simplify  the  following  fractions  and  check  each  result: 
,       ab^c"^  gg  -  3  r^  +  2 

_     21aj-10-9a;2 


bc^-J~a 
a'^-b^ 

a^'-b'^ 
x»  +  2/« 

x'  +  y^ 
x"  +  x^y 

£C*-2/2 

a'bcH'' 

-  ab'^cd'' 
mx^y  —  mxy^ 

6. 

£C*  —  y^ 

„       a^cH"" 

9-    TV-r.-  10. 


11.        -^Z        "Ta-  12. 

nx*y  —  nx^y^ 


3ic2-26x  +  35 

a«  +  g^  +  3  g  -  5 
^2  _  4  <^  +  3 

Q>x''  +  lxy-^y'' 
Q>  x^  +  11  xy-ir^y^ 

x^  -x^-lx  +  ^  ^ 
a;4  +  2ic«  +  2cc-l' 

x^  +  y''-z''  +  2xy^ 


FRACTIONS. 


119 


13. 


15. 


17. 


a^  +  a' 


2a 


23. 


a^  —  a^  —  6a 

3  x^y^  +  4  xy^ 
5  x^i/^  —  4  x^y 

a^'  +  Sa^lO 
Sa^  +  2a-16' 

x^  +  x^y  +  ^,y  ^ 

cc^  +  x^y^  +  icy* 

m«  -  39  m  +  70 
7?i2  _  3  ^^^  _  70 

x^  —  xy  —  12  7/^ 

ic^  +  5  ic?/  +  6  ?/^ 


14. 


16. 


18. 


20. 


22. 


24. 


x^  -\-x 

2-12  a^ 

ic^  +  4a;2 

+  5  a;  +  20 

1 

-6^2 

(1  +  axy 

'■-(a-^xy 

x'-5x' 

'-{-7X-8 

2^3 -5a 

;2_^4iC-l 

a« 

-a^x* 

a^  +  a^a?  - 

_  a'^x^  -  a^x^ 

77^^  —  6  m 

2  + 11m -6 

2  7/^3- 

14w  +  12 

ic^  +  (m 

—  7i)x  —  inn 

x  (x  -\-  m)  —  n  (x  -{-  7n) 


25. 


26. 


27. 


x^  -{-  (a  -\-  b)  X  -\-  ab 


(x  + 

a)(x 

+  h)(x 

+  <^) 

2a2_ 

-10  a- 

■28 

3a«- 

-27. 

7,2  +  21 

a +  147 

7?i2ic2 

-{m 

,  +  ?/)  m7?,a;  +  mn 

V 

;^  —  (771  +  1)  7ia;2  -f-  mn'^x 
[Omit  the  following  unless  Appendix  III  has  been  studied. 


28. 


29. 


30. 


31. 


(i\h  -c)  +  b^{c-a)  +  c^{a  -  b) 
abc  (a  —  b)(b  —  c)  (c  —  a) 

(a  —  b)  (b  —  c)  (c  —  a) 
a\b  -c)  +  b^{c-a)  +  c\a  -  b)  ' 

ab  (a  —  b)  +  be  (b  —  c) -[-  ca  (c  —  a) 
(a  —  b)  (b  —  c)  (c  —  a) 

ab  (a  -\-  b)-{-  be  (b  -{-  c) -\-  ca  (c  +  a) 
(a  -{-b)(b-\-  c)  (g  4-  a) 


120  ELEMENTS  OF  ALGEBRA. 


152.   Reduction  to  integral  or  mixed  expressions.     Since  the 

fraction  -  indicates  the  division   of   a   by  h^   it  may  be 

reduced  to  an  integral  form  if  the  division  is  exact,  and 
to  a  mixed  form  if  the  degree  of  the  numerator  equals  or 
exceeds  that  of  the  denominator  and  the  division  is  not 
exact. 

E.g., ^  =x  —  y,  the  division  being  exact. 

x  +  y 

x^  -\-  y^  2  2/^ 

=  X  —  y  -\ '■ —  ;  that  is,  the  division  of  the  remainder 

X  +  y  x-\-y 

hy  X  -{-  y  is  indicated. 

Check.     On  the  last  result.      Let  x  =  y  =  \.     Then  f  =  1  —  1  +  |. 


EXERCISES.    LXII. 

Reduce  the  fractions  in  exs.  1-10  to  integral  or  mixed 
expressions,  preferably  by  detaching  the  coefficients.  Check 
each  result. 

x'^  +  y^  _  x^  -\-  if  -{-  z^  —  ^  xyz 

1.  ,        '  2.    ■ 


x  +  y 

a^  +  3  «2  _  1 

a^  +  l 

3a;2  +  2ic-l 

3x-l 

^x'-^xy-^y^ 

x  +  y 

x^  -\-^  x^  -^Vlx 

+  8 

3.   — Vr-i 4. 


6. 


7.    .   -^    '   ^  •  8. 


9.     '■—: ■ 10. 


x^y  ^z 

4tx^-\-4.x^-^2x-\-l 
2x^  +  x-\-l 

x-^  -\- x^  +  x^  —  6 
x^  +  x^  +  x^  -i-  X  —  6 

a' -2  ab  +  52 
x^  +  4.x^y  4-  5xy^  +  2^ 


x  +  2  '  x^  +  3xy  +  2y^ 

11.    Show  that =  1  +  a  +  a2  +  ftS  +  a*  +  t^' 


FE  ACTIONS.  121 

153.  Reduction  to  equal  fractions  having  a  common  denomi- 
nator. 

Theorem.    If  Ti  T)  7  ^^^  ^^2/  fractions  whatever,  and  m 

is  any  coTwrnon  ^multiple  whatever  of  b,  d,  f,  it  is  possible  to 
reduce  the  given  fractions  to  equal  fractions  having  the 
commo7i  denominator  m. 

In  arithmetic,  for  example,  we  can  reduce  the  fractions 
h  h  \h  ^^  equal  fractions  having  for  their  common  denomi- 
nators 24,  48,  96  • .  •. 


Proof.    1. 

'.'  ??i  is  a  multiple  of  b, 
m  =  pb, 
m  =  qd, 
7n  ^=  rf 

d,f,  we  may  let 

2. 

a       pa      c        qc 
But  -  =  =^j     -j=     j' 
b      pb      d      qd 

and     ■^  =  ^-       §150 

3. 

a       pa      c  _qc 
'  '      b        m       d       711 

and     —  =  —■)   by    sub- 
f      m       ^ 

stituting  the  values  of  step  1. 

In  particular,  if  m  is  the  lowest  common  multiple  of  the 
denominators,  the  fractions  will  be  reduced  to  equal  frac- 
tions having  the  lowest  common  denominator,  a  step  of 
great  importance  in  working  with  fractions. 

E.q..  to  reduce  the  fractions and to  equal  fractions 

^'  X  -y  x  +  2/ 

having  the  lowest  common  denominator  : 

1.   The  l.c.m.  of  the  denominators  \s,  {x  +  y){x  —  y). 
2  a;  +  y  ^       {x  +  y)'^ 

Kx-y      (X  +  2/)  (X  -  2/) ' 
x-y {x  -  yf 


122  ELEMENTS   OF  ALGEBRA. 


EXERCISES.    LXIII. 


Reduce  the  following  to  equal  fractions  having  the  lowest 
common  denominator : 

X        y        z_^  ^     oh_      — ^'      a^ 

'    yz      zx      xy  '    cH      c^d}      de'^ 


3. 


y  z  xy  x'^y^ 


y  +  z      z  +  X      X  -\-  y  '    x^  -\-y^     x^  —  y^    ^  —  y 


X  +  \  X  —  1  X 


'    m^  +  6  ?/i  +  8      2  m^  +  7  m  +  6 

2m  —  2n  4  (w  +  n) 

7.    '     r~^ * 

m^  —  mn  +  7i^      5  (jn^  +  ^^  +  ^^) 

a^-P      (a  +  by      a^  +  b^  +  2ab 
^-    a^-b^'      a^  +  b^'         {a^-by 

9x^  +  12xy-5y^      6x^ -11  xy -]- 4^y\ 
^'     8x^-xy-10y^'      2x^-5xy  +  2y^' 

X  —  y  X  -\-  y  2  x^y'^ 


10. 


11, 


y{x-y)      x^  +  y{x  +  y)      x^  —  y^ 
2ic2  +  3cc-4      x^-2x^-8x  +  4. 


ic3  +  2a;2  +  3ic  +  4      x^-2x^-\-8x  +  4. 


io  ^  +  1  x  +  2  x^-8 

x^  +  5x  +  6     x''  +  4:X  +  3      a;2  +  3£c  +  2 

a  — 3  2ct  4-8  g  +  5 

*    a2-9a  +  18'    a2  +  a-12'    a^  +  Sa  +  ls' 

14.  ^y  2^5 ^  5^ 

(2/  +  ^)('^  +  x)     x^  -{-  zy  -\-  zx  -i-  xy    y'^  -\- yz  -\- xy -{- xz 


FRACTIONS.  123 


II.     ADDITION  AND   SUBTRACTION. 

154.  Theorem.  Operations  involving  the  addition  and  sub- 
traction of  fractions  can  be  performed  upon  the  numerators 
of  equal  fractions  having  a  common  denominator,  the  result 
being  divided  by  this  common  denominator. 

Proof.    1.  It  has  been  proved  in  §  87  that 
a       b        c  _a  -\-  b  -\-  c 

2.  .'.if  the  given  fractions  be  reduced  to  equal 
fractions  having  the  common  denominator  k, 
the  operations  can  be  performed  as  stated  in 
the  theorem. 

For  simplicity  it  is,  of  course,  better  to  reduce  to  equal 
fractions  having  the  lowest  common  denominator. 

Thus,  with  numerical  fractions, 

.2    _1_    5    _    4      1      5    _    9    _    3 

Examples.     1.  Kequired  the  sum  of 


b  —  c    b  -\-  c 

1.  The  l.c.m.  of  the  denominators  is  (6  +  c)  (6  —  c). 

2.  -^=       ^^  +  ^>^      ■  §150 
6  -  c      (6  +  c)  (6  -  c) 

o  a     _      {b  —  c)a 

6  +  c  ~  (6  +  c)  (&  -  c) 

4     .  _^  +  _^  =  (&  +  c)  g  +  (&  -  c)  g  ^^^ 

6  -  c      6  +  c  (6  +  c)  (6  -  c) 

2  ah 


{b  +  c){b-  c) 

Check.  If  a  =  1,  6  =  2,  c  =  1,  then  i  +  i  =  |.  It  is  not  permis- 
sible to  let  b  and  c  have  the  same  values,  because  that  would  make  the 
common  denominator  zero,  a  case  excluded  for  the  present. 


124  ELEMENTS  OE  ALGEBRA. 

X  X  -\-  S      X 2 

2.  Simplify  the  polynomial  ^^— ^  +  ^— -j  -  ^-^^  • 

1.    The  l.c.m.  of  the  denominators  is  x^  —  1. 

X  +  3  ^  (g  +  1)  (X  +  3) 
X  -  1  x2  -  1 

g  x-2^(x-l)(x-2) 

X  +  1  X2  -  1 

X  x  +  3      X -2  _x  + (x  + l)(x  + 3)-(x  -  l)(x -2) 

■  *  X2  -  1         X-1   ~X+  1   ~  X2-  1 

_x  +  x2  +  4x  +  8-x2  +  3x-2 

~  X2-1 

8x  +  1 


X2-1 

C/iecA;.     Let  x  =  2.     Then  2  +  s  _  o  _ 


155.    In  a  case  like  -^ ^ 7 ^?  it  must  be  remem- 

X''  -\-  y       ic^  +  y 
bered  that  the  bar  separating  numerator  and  denominator 
is  a  sign  of  aggregation. 

In  this  case  the  result  is  ^^V-J^-y)  =  ^^V-^^V  ^  _^JL_ . 

X2   +  2/2  a;2  +  y1  yp.  +  ^^2 


EXERCISES.    LXIV. 

Simplify  the  following  expressions,  checking  each  result 
by  the  substitution  of  such  arbitrary  values  as  do  not  make 
the  denominators  zero : 

1.    ^+A4._^.  2^"^^      ^~^ 


he      ca      ah  '    a  —  h       a  -{- h 

„      2x     ^     bz  ^     2-x    ,         x-2 


32/^2      g^2^  •    -j^_^2  '   \^x-2x^ 

-1,1  x  X  2xy 

x  +  y       x  —  y  '    x-\-y      x  —  y      x'^  —  if 


FRACTIONS. 


125 


g  +  l       Q^  +  2       a-1 
'    a  +  2       a-{-3      a-{-2' 

'    {x-yf      x  +  y      x-y 


9. 


10. 


5x^-1x^-^x^  +  11       x-1 

2x^-3x^  +  2x^-1       x  +  s' 


a  —  4:  a  — 5  a  — 3 


a' -9a +  20       a" -11  a       w"  -  1  a  +  12 


\.  1      _  2(1 -x)        1+^'   _  6x^(1  -x) 

^  '  1  +  x     {i+xy'^{i  +  xf      (i  +  xy 

a^  +  ab  +  b^      a^-ab  +  b'^      2b^  -b'^  +  a^ 
a  +  b  a-b        "*         a^  -  b^ 


L3. 


+ 


+ 


{a-b){a-c)       Q)-c)(b-a)       (c  -  a)  (c  -  b) 


14. 


+ 


+ 


(a  —  b)(a  —  c)       (b  —  c)(b  —  a)       (c  —  a){c  —  b) 
xy  yz  .  zx 


+ 


+ 


(y  -  ^)  (^  -  ^)      («  -x){x-  y)      {x  -y){y-  z) 
ax^  +  byz  ay'^  +  bzx  az^  +  bxy 


(x  -y)(x-  z)       {ij  -z){y-  x)       (z  -x)(z-  y) 

X      2x:'  +  7f      3  xy''  -3x^-y^      4:Xi/-2 xhf  -  ?/ 
y  xy  xy  xy^ 


L8. 


19. 


+ 


a{a  —  b)(a  —  c)       b(b  —  a)  (b  —  c)       g(c  —  a) (c  —  b) 
xy  yz  zx 


{y  +  z){z  +  x)    '    {z  +  x)  (x  +  y)       (x  +  y)  {y  +  z) 
2xyz 

(^  +  y){y  +  ^)  («  +  X) 


126  ELEMENTS   OF  ALGEBRA. 

III.     MULTIPLICATION. 

156.  Theorem.  The  product  of  two  fractions  is  a  frac- 
tion whose  numerator  is  the  product  of  their  numerators 
and  whose  denominator  is  the  product  of  their  denomina- 
tors. 

Given         the  two  fractions  -z^  -' 

0    d 

To  prove    that 
Proof.    1.  Let 

2.  Then  bdx  =  b-~-d--^  Ax.  6 


a   c 
Td 

_  ac 
^bd 

X 

a   c 
^b'd' 

bdx 

=  b---d-- 
b        d 

=  ac,  for  b 

a 
~b 

X 

ac 
~bd 

a    G 
b'd 

ac 
~bd 

3.  =  ac,  for  b  --  =  a,  by  def .  of 

division 

4.  .'.  ^  =  F7*  ^x.  7 
bd 

^  a    G       ac  ^      ^ 

b   d      bd 

Corollaries.     1.   Similarly  for  the  product  of  any  num- 
ber of  fractions. 

c       ao 
2.   The  product  a--  =  —  j  as  defined  in  §  52. 

For  iih  =  \,  the  identity  -  .  ^  =  ^  becomes  «•-  =  -• 
b   d      bd  d       d 

Illustrative  problem,     -z -z^ — ^— -—  •  — — — 

^  x^-12x-\-35    x^ -11x^12 

_{x-h){x-Z){x-l){x-^)  _x-?> 

~  (03  -  5)  (ic  -  7)  (£c  -  9)  (ic  -  8)  ""  ic  -  9* 

^,     ,       8    42       -2 
Check. = 

24    56       -8 


FRACTIONS.  .  127 


EXERCISES.    LXV. 


Perform  the  multiplications  indicated,   simplifying  the 
results  and  checking  as  usual. 


7  x^y'^     18  xHf  x^  —  if    x^  —  xy  -\-  y 

^'    12  xy*  28icy'  '    x^  ^if'  x'^xy  -\-  y 

21 X         x  +  y  a^  +  b''  +  2ab         1 


8y  +  8x        3  a-b  a'-b^ 

x^-y^     x^-y"^  (aJ^b){x  +  y)  a'^-b'^ 

x^  —  if    (x  +  yY  '    (a  —  b)(x  —  y)  x  +  y 

cc*  —  ?/*       ^  —  y  ^ 


(x  —  yy    x^  -\-  xy    x^  -\-  y^ 

X*  +  x^y  +  xy^  +  y^     ^  —  y  _ 

x^  -{-  2  xy  -\-  y'^         x^  -\-  xy 

g;^  +  ^  -  12      g;^  +  2  a;  -  35 
^*    cc'-^-13ic  +  40*  cc2_^9x  +  20 

ic2  +  5  a;  +  6      cc'^  +  9  ic  +  20 


10. 


11. 


a;2 +  7x4-12    ^2  + 11^^  +  30 

2  ft^  +  5  ^  +  2    9a^  +  15a  +  4 
6a2^5a  +  l*  5a2  +  12a  +  4 


Reduction  of  integral  or  mixed  expressions  to  fractional  form. 

157.    Theorem.     An  integer  can  always  be  expressed  as  a 
iction  with  any  denominator. 

For  since  1  =  7' 


128 


ELEMENTS  OF  ALGEBRA. 


158.   Theorem.    A  mixed  expression  can  always  be  written 

in  fractional  form. 

h      ac      b 
For  since  «  +  -= 1 —  -        §  157 


b      ac  -{-  b 

a-{--  = 

c  c 


§154 


EXERCISES.    LXVI. 

Write  the  expressions  in  exs.  1-8  as  fractions  with  the 
denominators  indicated,  as  in  §  157. 


1. 

5,                      denominator  25  a. 

2. 

abc,                                      ' 

'     abc. 

3. 

^  +  y,                        ' 

'     a? -2/. 

4. 

x'  +  x'-{-x  +  l, 

'     x-1. 

5. 

x^-x^  +  x^-x  +  1,       ' 

'     x  +  1. 

6. 

a«  -  b% 

'     a^  +  bK 

7. 

x^-j-xy-^  y\ 

'     x^-xy  +  y\ 

8. 

{a-b){b-c){c-a),     ^ 

i     (a  +  b){b-\-i 

Reduce  the  following  to  fractional  forms,  as  in  §  158, 
checking  each  result : 


9.   4.  a 


6ab-2 
Sb 


11.     CC^  +  iC  +  1  + 


x-1 
13.    1  +  a  +  a2 


10.    a  +  b-i 

a  —  b 

12.   ^8_3^_3a;(3-a;) 

x-2 


a^  + 
14.    x^ -{- 2  xy -{- 2/ - 


a-1 

(x^  -  yy 


x^  —  2xy  -\-  y^ 


FRACTIONS.  129 

159.  Theorem.  Any  integral  power  of  a  fraction  equals 
that  poiuer  of  the  numerator  divided  by  that  power  of  the 
denominator. 

Given         the  fraction  y?  and  the  integer  n. 
To  prove    that  (  ^  )  =  i"  * 


/aV_  a    a    a 


Proof.    1.  I  T  1  =  T  •  T  •  T  •  •  •  to  TO  factors 

Def.  of  power 

_  aaa ■■  •  to  n  factors       „  ^^^  ^ 

2.  =  —7 ^— §  156,  cor.  1 

bob  •  • .  to  TO  lactors 

3.  —J^'  Def.  of  power 


EXERCISES,    LXVII. 

Express  the  quantities  in   exs.   1-6  without  using  the 
parentheses.     Check  each  result. 


\'    \^  —  yj  '    \a  —  by  '    \x  —  3 

(a  +  Z>  +  c V  m'^  -\-m-\-l  fp-\-q-\-f\ 

abc      J  '       (jn  +  iy  '    \p—q—r) 


Express  the  following  quantities  as  powers  of  a  fraction  : 

ft2  ^_  lOaJ  +  25Z>2  a*  +  9/>^  +  ^a%'' 

8. 


1  +  4  ic  +  4  £c2 

100£C4  +  20£C2  +  1 

a;4  +  20ic2^100 

4  a;2  4-  9  ?/2  +  12  aj?/ 

81(a2  +  ^;2)  +  162a^> 
a;6^X+3r«^(a;^  +  l) 

11     ---     .    -..     ,   — -^  a;3  _  3  a;2y  4-  3  a;y^  -  y^ 

■    4ic2-f  92/'-12a;y/  '       a;«  +  3  ic^  +  3x  +  1 


130  ELEMENTS  OF  ALGEBRA. 

Illustrative  problems   in   multiplication.      1.    To  find  the 

,        '.  X  —  a    X  —  2a         ^       x 

product  01  >  -p—i  and 

■^  X -[- a    x-\-za  x  —  a 

-1     -o    Pie^^~<*^"~2a       X     _       (x  —  a)(x  —  2a)x 

1.  liy  §  loo, • • =: — 

X  +  a  x  +  2a  x  —  a      (x -\-  a){x +  2a){x  —  a) 

(x  +  a)  (X  +  2  a) 

2    13         3 

Check.     Let  x  =  3,  a  =  1.     Then = 

4    5    2      4-5 

2.  To  find  the  product  of  -  +  -  and 

0      a  0      a 

/a  .  6\/a      h\     /a\^     /b\'^ 


{i-i)Q^i>Qr-& 


2.  ^'^-^  §169 

C/iecA:.     Let  a  =  1,  6  =  2.     Then 

\2/V2         /  4'  22  4 

3.  To  find  the  product  of  -^ = -zrx  •  —z z r ' 

x2  +  6x  +  5    x2  +  8x  +  15 


x2  +  7x+12    x2  +  5x  +  4 

_(x  +  l)(x  +  5)    (X  +  3)  (X  +  5) 
~  (X  +  3)  (X  +  4)  ■  (X  +  1)  (X  +  4) 


,  by  factoring 


(X  +  4)2 
12    24      36 


_(x  +  l)(x  +  5)(x  +  3)(x  +  5)  ^gg 

(x  +  3)(x  +  4)(x  +  l)(x  +  4) 


Check. 

20    10      25 


I 


7.    1 


FRACTIONS.  131 

EXERCISES.    LXVIII. 

Perform   the   multiplications    indicated,    simplify    each 
result,  and  check. 

■i(-f)('-^)- 

ri       2x   "I  rs _    3x   "I 

a  -{-  h  f      a  a  —  b       a  —  b\ 

a  —  h  \a  -\-  h  a  a  -\-  h  J 

\a       bjG      \a       cj  b       \b       cj  a 

9    A  I  1  ,    ^  +  M    ^    ^^  +  ^^    Y 

10  ^  /^i-^i^  2  A    ,    A_J:_ 

•    (a  +  ^,)2  *  \^«2  +  ^2^  "^  (a  +  ^,)3  •  y^  ^  b)~  a'b''' 

I-'+(i+9(M)(^;> 


^*  a;2-2aic  +  a'  x' -2(b  +  c)x+(b  +  cy 

f         /  a;^  a;«y        xY      4a;?/^       16  y^\    /  a;         2y\ 

•    Vl6^'      12^«"^9«*       27^2"^    81  y    V2«'       3/ 


132  ELEMENTS   OF  ALGEBRA. 


IV.     DIVISION. 


160.  The  fraction  formed  by  interchanging  the  numer- 
ator and  denominator  of  a  fraction  (of  which  neither  term 
is  zero)  is  called  the  reciprocal  of  that  fraction. 

E.g.,  2  Ls  the  reciprocal  of  -,  -  is  the  reciprocal  of  -,  and  -  is 
^,  .  ,    .  a  2    3  2  a 

the  reciprocal  of  -  • 

Evidently  1  and  —  1  are  the  only  numbers  which  are  their  own 
reciprocals,  respectively. 

The  term  reciprocal  is  used  only  in  relation  to  abstract  numbers. 

161.  Theorem.  To  divide  any  number  by  a  fraction  is 
equivalent  to  multiplying  that  number  by  the  reciprocal 
of  the  fraction. 

Given         the  fraction  -  and  the  number  a. 

b  ^ 

To  prove    that  q  -^-7^  -  ■  q- 
0       a 

Proof.    1.  Let  x  =  q-T---  ■ 

2.  .'.        J  ■  X  =  q,  hj  def.  of  division,  or      Ax.  6 

^         b    a  b         ^  ,     ,       ^ 

3.  .'.-•-•  X  ='-  •  q,  by  mult,  by  - •  Ax.  6 

a    b  a    ^'     -^  -^    a 

4.  .-.  x  =  -'q,  since  -'7  =  1.    §§156,150 

a  a    b 

O.  .  .       q  -i--  =  -  '  q.  Ax.  1 

b       a    ^ 

162.  Corollaries.  1.  The  reciprocal  of  a  fraction 
equals  1  divided  by  the  fraction. 

For  1  -i.     =  _  .  1^  by  the  theorem. 

0      a 

a  +  b  +  c      1         ,    1    ,  _L  1 

^.    =  —  -aH bH c. 

m  m  m  m 

For  to  divide  by  m  is  to  multiply  by  its  reciprocal. 


FRACTIONS. 


133 


Illustrative  problems.     1.  Perform  the  following  division : 
27  3x 


S(x'-y^   '  x-y 

1. 

27 
8(x2-2/2) 

Sx   _x  -  y              27 
x-y        3  X     8  {X -{- y)  {X  -  y) 

§161 

o 

{x-y)27 

§156 

Sx-S{x-i-y){x-y) 

3. 

= ,  cancellinff  3  (x  - 

8x(x  +  2/)'                  ^     ^ 

-y)- 

§150 

.  ..      .     ^u„„   27        6          9       ,_ 

9      , 

.       3 

Check.     Let  x  =  2,y  =  l.    Then =-  -  = ,  f  or  -  -^  6  = 

8-3      1       16.3  8  16 


).  Perform  the  following  division 


X  -\-  a       x^  +  a^ 
X-  a      x^  -{-  a^  x^-  a^ 


x  +  a       x3  +  a3      X  —  a     x  +  a 

_  (x3  +  a^)  (x3  -  a^) 
~    (x  —  a)  (x  +  a) 
=  (x2  -xa  +  a2)  (x2  +  xa  +  a^). 
Check.    Let  x  =  2,  a  =  1.     Then 

^-^  -  ^^^  =  (4  -  2  +  1)  (4  +  2  +  1),  for  -  -f-  -  =  21. 
2  +  1      8  +  1      ^  n-MTT-^         g      Q 


§161 
§156 


3.  Perform  the  following  division  : 

b     \       w'  +  h' 


a  —  b       a  -\-b 


a^  —  ab 


a2  +  62 


a  —  h      a  +  6     (a  —  6)  (a  +  6) 
a2  +  62  a2  +  62  _       a2  ^  62         a  (a  -  6) 


(a  -  6)  (a  +  6)      a2  -  a6     (a  -  6)  (a  +  6)     a2  +  62 
a 


161 


a  +  6 


C/iecfc.     Let  a  =  2,  6  =  1.     Then  f -f- f  =  |. 


iB4  ELEMENTS  OF  ALGfiBtlA. 

EXERCISES.    LXIX. 

Perform  the  following  divisions,  simplifying  each  result, 
and  checking. 

^'    a^  +  Sa- 33  '  a' +  7  a  -  u' 


x^  -\-  X  (a  -^  b) -{-  ab   _  /"x  -\-  a\ 

'•     x^-\-x{b  +  c')-^-  be    '  \x-\-cJ 

ai^^2_^2_^2ab  {a  +  b  -  cf 
^'          (a  +  b  -\'  cy         '  abc 


25x-29 


\5-4:X      2-x)      (5-4ic)(2-a;) 

fa  a     \       I      a  1  —  a\ 

'     \l-\-a      l  —  a)^\l  +  a  «""/ 

(X  x  —  l\       f     x        I   ^  —  l\ 

iC  +  l  X      J    '    \x  +  1  X      ) 

fa"  +  b''  _  w"  -  b'\  ^  U^b  _  a-b\ 
•    ^2-^.2      a'^^b'')   ''\a-b       a  +  bj 

(la-\3 
\    a -3b 


3.    .   ■--13^^2^^-5^_^, 


b    ^   3b-a  ^J    '  a-3b 

x'^-Q>xy  +  9i/  ^  fx^  —  9y^  ^  x^ -{- xy  -  Q  y^\ 
•    x^-^xy +  4.y^  '   \x''  +  4:y^^  x''-xy-Q»y^)' 

10.  f^^+_3^y  3.-^^/_l i_\ 

\3a-b      3a-bJ    da'^  +  b^      \3  a  -  b      3a^b) 
'    m  +  n   '  \  W(a  +  b)     '  |_    5x^y      '  7 (m^ -  72:')  j 


FRACTIONS.  135 

V.     COMPLEX   FRACTIONS. 

163.  A  fraction  whose  numerator,  denominator,  or  both, 
fractional  is  called  a  complex  fraction. 

a  +  6  a  +  b 

E.g.,  ,  ,  are  complex  fractions. 

^  '  a^  ^  ab  +  b'^'  b_+_c'  oj-b  ^ 

be  a 

164.  Complex  fractions  are  simplified  either  by  perform- 
[g  the  division  indicated,  or  by  multiplying  both  terms  by 

such  a  factor  as  shall  render  them  integral. 

a  4-  6 
c      _      c^       a  +  b  §  150 


E.g., 


§§  150,  156 


c2 

_     c 
~  a  —  b 

a  +  b 


Or,  =  — ,  by  multiplying  both  terms  by  c2, 

a2  —  62       ^2  —  62 


= ,  by  cancelling  a  +  b. 

a  —  b 

Check.     Let  a  =  2,  6  =  1,  c  =  1.     Then  f  =  1. 

It  is  obvious  that  the  latter  plan  is  the  better  when  the 
multiplying  factor  is  easily  seen. 

X2-2/2 


E.g.,  to  simplify  — 


y1 

Multiplying  both  terms  by  xy2,  this  equals 
y  (g2  -  y^)  _  y  (x  +  y) 
x(x-y)  ~       X 

Check.     Let  X  =  2,  y  =  1.     Then  f  =  |. 


136  ELEMENTS   OF  ALGEBRA. 

EXERCISES.    LXX. 
a  +  b 
1.    Simplify  -. 


a  +  b 

2 


^  -y 

2.    Simplify 


X 


x  +  y 


y 

a-b        1 


3.    Simplify  - 


5a  +  lb 
a  -\-l      a  —  1 


4.    Simplify  ""-]       ^  +  ^ 


a  —  1       a  +  1 

5.  Simplify ;+-:  1+-:. 

1  +  a^  ~"  l  +  a* 
6.    Simplify  ^(^~^)-H<^  +  b) 


a  +  b       a  —  b 


7.    Simplify  the  reciprocal  of  "'  "^  ^^ 


a 


a^  +  b^ 


8.    Simplify  the  reciprocal  of  — — 

9  b      15  a 


i 


FRACTIONS. 


137 


Simplify  the  following  expressions  : 
a 


1  + 


9. 


10. 


1  +  «    .    («  +  1)^ 


a  + 


a^  +  a  +  1 


1-^a 


(a  +  by  —  a^  —  b^     a        b 

{a  +  bf  -a^-b^'  a^  _b2 

b        a 


11. 


12. 


a^  +  a%''  +  Z**      a^-  b' 


1  .         ^.       a  —  b 

'  (a  +  b) 


a^'  +  h 

1  + 


7  ~r        9 


*1 


b^ 


13. 


('-d('-;)'('-;)('-5) 


+ 


('-3('-d 


14.     ^ 


x^       y^       x^       y^ 


x^       y^       x^       y^ 


f^  +  y 

\x-y 


_^x-y\fx^       y^ 


-) 


138  ELEMENTS  OF  ALGEBRA. 

165.   Continued  fractions.     Complex  fractions  of  the  form 


b  + 


d  + 


are  called  continued  fractions. 

Such  fractions  are  usually  simplified  to  the  best  advantage 
by  first  multiplying  the  terms  of  the  last  fraction  of  the 

form by  the  last  denominator,  /,  and  so  working  up. 

E.g.,  to  simplify  the  fraction  - 


1 


Multiplying  the  terms  of by  c,  the  original  fraction  reduces 

1 

to Multiplying  the  terms  of  this  fraction  by  &c  +  1,  this 


reduces  to 


6c +  1 

6c +  1 


a6c  +  a  -{-  c 

1  2 

Check.     Let  a  =  6  =  c  =  1.     Then =  -■ 

1  +i      3 


EXERCISES.    LXXI. 


a 

1. 

Simplify 

0 

1  + 

1-^: 

a^ 

2 

Simplify 

1 

X  — 

1 

X  — ■ 

1-x 


FRACTIONS.  139 


3.    Simplify  ^ 


1 


4.    Simplify   — 


X 


.+  1 


5.    Simplify 


6.    Simplify 


7.    Simplify 


x-1 
1 


x^ 


x+      1 


X  -\-l 

X  +  y 


x  +  y  -\ 


x-y  + 


x  +  y 
a 


a 


l  +  a+  '' 


8.    Simplify  a'  + 


1  +  a  +  ^2 

2 


a^+  ' 


a^  — ; 


9.    Simplify  x  -{-  y  -{- 


a" 
1 


x  -\-  y  -\ 


1 
x  +  y  + 


10.    Simplify  (a  +  hf  ^ 

(a  -  by  + 


x  +  y 
1 


(a  +  by  + 


1 

(a-b) 


140  ELEMENTS   OF  ALGEBRA. 

VL    FRACTIONS  OF  THE  FORM  ^>   ^>   AND  |-- 

166.  By  the  definition  of  fraction  (§  149)  expressions 
of  division  in  which  the  divisor  (denominator)  is  zero 
were  excluded.  An  interpretation  of  this  exceptional  case 
will  now  be  considered. 

When  the  absolute  value  of  a  variable  quantity  can 
exceed  any  given  positive  number,  the  quantity  is  said  to 
increase  without  limit,  or  indefinitely. 

E.g.,  in  the  series  -,  — ,  — r-,  ,  •  •  •,  the  values  of  the  suc- 
cessive terms  are  1,  10,  100,  •  •  • .  Hence,  as  the  absolute  values  of  the 
denominators  are  getting  smaller,  the  absolute  values  of  the  fractions 
are  getting  larger  and  may  be  made  to  increase  without  limit. 

The  symbol  for  an  infinitely  great  quantity  is  oo,  read 
'*  infinity."  This  symbol  must  not,  however,  be  understood 
to  have  a  definite  numerical  meaning.  It  is  merely  an 
abbreviation  for  "a  quantity  whose  absolute  value  has 
increased  beyond  any  assignable  limit." 

Hence,  go  +  a  =  oo. 


and  —  =  00. 

a 

In  fact,  the  symbol  oo  is  not  subject  to  any  of  the  common  laws  of 
numbers. 

167.   If  a  is  a  constant  finite  quantity,  the  absolute  value 

of  -  can  be  made  as  small  as  we  please  by  increasing  x 

sufficiently.      That  is,  -  can  be  brought  as  near  0  as  we 

please.    This  is  expressed  by  saying  that  the  limit  of  ->  as 
X  increases  indefinitely,  is  0. 

This  is  written,  -  ==  0  as  x  increases  without  limit,  the  symbol  == 


being  read  "  approaches  as  its  limit,' 


FRACTIONS.  141 


0  x^  —  a'^ 

168.   The  form  -  •    The  fraction has  a  meaninsr  for 

0  X  —  a  ^ 

^2    qJI  rj^    Q^ 

all  values  of  x  except  x  =  a.    But = (x  -\-  a), 

X  —  a       X  —  a    ^  ^ 

and  as  ic  =  ct  it  is  evident  that (x  -\-  a)  =  1  •  (a  -{-  a) 

X  —  a    ^  ^  ^  ■^ 

—  2  a. 


\ 


x^  —  1     X  —  1 

Similarly,         = (x  +  1),  which  =  2  as  x  =  1 

X  —  1      X  —  1 

X2— 4x  +  4        X  —  2.  „.  ,.    u.r^  .o 

—  (x  —  2),  which  =bO  asx  =  2  ; 


x-2  x-2 

X^  -  1  _  X  -  1 

X  —  1  ~  X  —  1 


(x2  +  X  +  1),  which  =  3  as  X  =  1. 


But  all  these  fractions  approach  the  form  ^  as  cc  ap- 
proaches the  limit  assigned,  and  in  the  several  cases  the 
fractions  approach  different  limits.  And  since  the  limits 
are  undetermined  at  first  sight,  §  is  said  to  stand  for  an 
undetermined  expressiori. 

This  is  commonly  expressed  by  saying  that  §  is    indeterminate. 
le  limit,  however,  can  often  be  determined  by  simple  inspection. 

x^  —  1   . 
[169.   The  fact  that  the  limit  of  —  is  2  as  a?  =  1  is 

)ressed  in  symbols  thus  : 


x-lji 


EXERCISES.    LXXII. 

Find  the  limit  of  each  of  the  following  expressions : 

,8 


1.  ^ 


I 


x  — 


i1  .  2.    El^2-| 


142  ELEMENTS  OF  ALGEBRA. 

£c2  +  2a;-8" 


X 


2a;-8"[  x^ -^x" -hx -\- ^'^ 

-2   J;  ^*     x'-^x^?,   J; 

a;^  +  2a;-8"[     ^  x""  -^  2  x'' -\- 2  x  ^  \^ 


170.  The  form  -•  This  form  should  be  interpreted  to 
mean  an  expression  whose  absolute  value  is  infinite. 

For  in  the  fraction  -  ?  as  x  ==  0  the  absolute  value  of  the 

X 

fraction  increases  without  limit. 

Hence,  the  symbol  -j  while  without  meaning  by  the  com- 
mon notion  of  division,  is  interpreted  to  mean  infinity. 

171.  The  form  —  •  This  form  should  be  interpreted  to 
mean  an  expression  whose  absolute  value  is  zero. 

For  as  x  increases  without  limit,  -  =  0. 

X 

172.  The  form  oo  •  0.  This  form  should  be  interpreted  to 
mean  an  undetermined  (indeterminate)  expression.  (Why  ?) 

173.  The  relation  of  these  forms  to  checks.  The  student 
has  been  cautioned  against  substituting  any  arbitrary  values 
which  make  the  denominator  of  a  fraction  zero.  The  reason 
is  now  apparent. 

12  1  1 

-E'.cr., = If  X  =  1,  this  reduces  to  oo  —  oo  =  -, 

which  checks  nothing  because  oo  has  no  exact  numerical  value. 
Similarly  in  the  case  of 

a  -  62      a2  -  6  ~  (a  -  62)  {g?.  _  6) 

If  a  =  6  =  1,  this  reduces  to  §  +  ^  =  §,  which  checks  nothing  because 
§  has  no  exact  numerical  value.  But  if  a  =  2  and  6  =  1,  this  reduces 
to  3  +  i  =  V-. 


FRACTIONS.  143 


EXERCISES.     LXXIII. 

1.  How  should  -  be  interpreted  ?     Why  ? 

2.  Also  |. 

3.  Find  the  limit  of : • 

.  n.   a, b G 

(a  —  b)(a  —  c)^  (b  ~  c)(b  —  a)     (c  —  a)(c  —  b) 
What  arbitrary  values  are  excluded  in  the  check  ?     Why  ? 


/T  A  fin  (1 

5.  Similarly  with  — h  — +  — -•     Why  is  it 

•^  b  a^  —  b^      a^  —  b^  ^ 

no  check  to  let  a  =  ^  =  any  number  ? 

6.  Similarly  with 
1      ^  _1_  ^  _i_  ^  {a-bY^{b-cY^{c-aY 

a  —  b      b  —  G      c  —  a  2(a  —  b)  (b  —  g)  (g  —  a) 

7.  Similarly  with 

r       a b  2b^  -\    a_ 

\_{a  +  bf      a'-b'''^  {a  +  bf{a  -  b)  \'  a 

8.  Show  that 
1  1  2a  2a' 


+  b 


1-a       1  +  a       1  +  a'^       (1  -  «)  (1  +  a2^ 

2a^ 


(1  +  a)  (1  +  a^)       1-6^4 

9.    Verify  the  following  identity,  (1)  by  actually  adding, 
(2)  by  the  substitution  of  arbitrary  values. 
xYz^      (x^-b^)(y^-b'')(z^-b^)      (x^-G^)(y^-G^)(z^-G^) 

bH^    ^  b'^QP'-C^  G^{f-b^) 

=  X-  +  y^  +  z'^-b^-  c\ 


144  ELEMENTS   OF  ALGEBEA. 


REVIEW  EXERCISES.    LXXIV. 

1.  What  is  the  value  of  -  + -,  when  x  = 

a      b~a       a+b 

b{b  +  a)   ' 

2.  Show  why  the  arithmetic  definition  of  fraction  is  not 
sufficient  for  algebra. 


3. 


Simplify  the  expression  -  —  -7-  +  y(-  +  l) 


4.  Extract  the  square  root  of 

(a)  9x74-3ic*  +  lli«V5-4/5  +  4/25iz;2. 

(b)  1  +-  4/cc  +  20/£c8  +  25/^*  +  lO/ic^  +  24/^^  +  16/a^«. 

(c)  178/7-20£c/72/  +  9?/V16£c2+4a;V497/2-15y/2(r. 

5.  Extract  the  cube  root  of 

(a)  8xVa3  +  48ic7a2_|_96ic/rt  +  64. 

(b)  \o.^-l  a%  -  ^^b^  +  I  a^c  +  1  aZ»2  _^  1  Z»2c  +  1  c8  +  3  ac" 
—  \bc'^  —  \  abc. 

(c)  l-3cc/2  +  3icV2-5:rV4  +  3icV4-3icV8  + 
5ic732  -  3a;V64  +  3  0^7256  -  ^V  5 12. 

6.  Prove  that  the  sum  of  two  quantities,  divided  by  the 
sum  of  their  reciprocals,  equals  the  product  of  the  quanti- 
ties. 

7.  Show  that  by  substituting  3  (cc  +  1)  /(cc  —  3)  for  x  in 
either  of  the  expressions  (3  —  4  cc  +  x^)  /  (3  +  x'^),  2  (3  +  a^)  / 
(3  +  x"^),  it  becomes  identical  with  the  other. 

8.  Eaise to  the  fourth  power.     Check. 

9.  Raise  —  -[- -^^  to  the  sixth  power.     Check. 


CHAPTER   IX. 

SIMPLE   EQUATIONS   INVOLVING   ONE   UNKNOWN 
QUANTITY. 

I.     GENERAL  LAWS   GOVERNING  THE  SOLUTION. 

174.  An  equation  has  already  been  defined  (§  16)  as  an 
equality  which  exists  only  for  particular  values  of  certain 
letters,  called  the  unknown  quantities. 

E.g.^  x2  =  4  exists  only  for  the  two  values  x  =  +  2  and  x  =  —  2. 

175.  An  equation  is  said  to  be  rational,  irrational,  integral, 
or  fractional,  according  as  the  two  members,  when  like  terms 
are  united,  are  composed  of  expressions  which  are  rational, 
irrational  (or  partly  so),  integral,  or  fractional  (or  partly 
so),  respectively,  with  respect  to  the  unknown  quantities. 

E.g.^         X  +  Vs  =  0  is  a  rational  integral  equation  ; 

5  +  i  vx  =  0  is  an  irrational  integral  equation ; 

-  +  4  =  X  is  a  rational  fractional  equation ; 

X 

1 

4  is  an  irrational  fractional  equation. 


(X  +  2)^ 

176.  A  rational  integral  equation  which,  when  its  like 
terms  are  united,  contains  no  term  of  degree  higher  than 
the  first  with  respect  to  the  unknown  quantities,  is  called 
a  simple  or  a  linear  equation. 

E.g.^  X  —  3  =  5,  x2  +  X  —  1  =  x2  +  2/,  are  simple  or  linear  equations. 
But  Vx  =  5,  -  =  2,  are  not  as  they  now  stand. 

145 


146  ELEMENTS  OF  ALGEBRA. 

177.  Equations  which  are  not  simple  are,  however,  often 
solved  by  the  principles  which  govern  the  solution  of  simple 
equations. 

E.g.,  (x  —  l)(x  —  2)  =  0  is  an  equation  of  the  second  degree.  (Why?) 
But  it  is  satisfied  only  if 

X  -  1  =  0, 

or  if  X  -  2  =  0, 

that  is,  if  X  =  1,  or  if  X  =  2.     Hence,  the  solution  of  this  equation  of 

the  second  degree  reduces  to  the  solution  of  two  linear  equations. 

EXERCISES.    LXXV. 

1.  What  is  meant  by  the  roots  of  an  equation  ?  (See  §  16.) 
What  are  the  two  roots  of  the  equation  x^  =  25? 

2.  What  is  meant  by  solving  an  equation  ?  Solve  the 
equations. 

(a)  3ic  +  5  =  0.  (b)  (x-2)(x-3)=0. 

(c)  (x  +  l)(x  +  2)=0.     (d)  (x  +  2)(x-3)=0. 

3.  What  is  meant  by  an  equation  being  satisfied  ?  What 
values  of  x  satisfy  these  equations  ? 

(a)  (x  +  ^)(3x-2)=0. 

(b)  (2£c-l)(2ic  +  3)=0. 

(c)  x(x-l)(x-  2)  (ic  -  3)  =  0. 

4.  What  is  meant  by  the  members  of  an  equation  ?  How 
do  they  differ  from  the  terms  ? 

5.  Which  of  the  following  are  simple  equations  with 
respect  to  cc  ? 

(a)  x^  -{-  x^  -^  X  —  x^  —  x^  =  A. 

(b)  3x^  +  x  +  7  =  2x^-^x(x  +  3). 

(c)  V^  +  4  =  7.  (d)  i-^  =  3. 

^  '^  ^  ^  X      2 

(e)  x^-x  +  l=:0.  (f)  x(x  +  l)=^x^ 


¥ 


SIMPLE   EQUATIONS.  147 

178.  Known  and  unknown  quantities.  It  is  the  custom 
to  represent  the  unknow7i  quantities  in  an  equation  by  the 
last  letters  of  the  alphabet,  particularly  by  x,  y,  z. 

This  custom  dates  from  Descartes,  1637. 

179.  Quantities  whose  values  are  supposed  to  be  known 
are  generally  represented  by  tlie^rs^  letters  of  the  alphabet, 
as  by  a,  ^,  c,  •  •  • . 

E.g.,  in  the  equation  ax  +  6  =  0,  a  and  6  are  supposed  to  be 
known.  Dividing  both  members  by  a,  x  +  h/a  =  0,  which  is  satisfied 
if  X  =  —  b/a. 

180.  The  solution  of  the  simple  equation  has  already 
been  explained  (§  17).  The  general  case,  not  involving 
fractional  coefficients,  will  be  understood  from  two  illustra- 
tive problems  and  the  series  of  questions  in  the  following 
exercises. 

1.  Given  the  equation  5ic  — 2  =  3ic  +  8,  to  find  the 
value  of  X. 


1.    5x-2  =  3x  +  8. 

Given. 

2.           5x  =  3x  +  10. 

Adding  2.                       Ax.  2 

3.           2x  =  10. 

Subtracting  3  x.              Ax.  3 

4.             x  =  6. 

Dividing  by  2.                Ax.  7 

Check.     Substitute  5  for  x  in  the  original  equation,  and 

25  -  2  =  ] 

L5  +  8. 

2.  Given  the  equation  2  ax  - 

-  a^  =  ax  -\-3  a^,  to  find  the 

value  of  x. 

1.  2  ax  -  a2  ==  ax  +  3  a^. 

Given. 

2.            2  ax  =  ax  +  4  a^. 

Adding  a'^.                      Ax.  2 

3.              ax  =  4  a\ 

Subtracting  ax.              Ax.  2 

4.                x  =  4a. 

Dividing  by  a.                Ax.  7 

Check.     Substitute  4  a  for  x  in  the  original  equation,  and 

8  a2  -  a^  =  4 

a2  +  3  a2. 

148  ELEMENTS   OF  ALGEBRA. 

181.  From  these  illustrative  problems  it  will  be  observed 
that  any  term  may  be  transferred  from  one  member  of  an 
equation  to  the  other  if  its  sign  is  changed.  This  opera- 
tion is  called  transposition. 

E.g..,  if  ic  +  2  =  7,  transposing  2  we  have 

a  =  7  -  2, 
or  x  =  5. 

It  should  be  remembered,  however,  that  the  operation  is  really  one 
of  subtraction,  2  being  taken  from  each  member  by  ax.  3. 

In  general,  transposition  is  always  an  operation  of  subtraction  or 
addition. 

EXERCISES.    LXXVI. 

The  answers  to  the  following  questions  will  lead  to  the 
understanding  of  the  steps  to  be  taken  in  the  solution  of 
linear  equations  with  one  unknown  quantity. 

1.  In  which  member  do  you  seek  to  place  the  known 
quantities,  and  in  which  the  unknown  ?  Might  this  be 
changed  about  ?  What  axioms  are  involved  in  this  opera- 
tion ?     (See  ex.  1,  steps  2  and  3,  p.  147.) 

2.  Having  done  this,  what  is  the  next  operation  ?  What 
axiom  is  involved  ?     (See  ex.  1,  step  4,  p.  147.) 

3.  State,  then,  the  two  general  steps  to  be  followed  in 
solving  a  linear  equation  with  one  unknown  quantity. 

4.  How  is  the  work  checked  ? 

Solve  the  following  equations,  checking  the  results. 

5.  12a;-28  =  8-f3ic.  6.    11  -x  =  2x-l. 
7.    27a; -127  =  11 -19a;.          8.    2a; -f  3  =  4ic -f  5. 

9.    4a; -34  =  22 -3a;. 

10.  a;-f2  +  3a;-}-4  =  5a;  +  6. 

11.  3a;  +  4a;  +  5a;  =  6a; +  72. 


SIMPLE   EQUATIONS.  149 

182.  The  axioms  applied  to  the  solution  of  equations.  While 
it  is  true  that  the  solutions  of  equations  depend  upon  cer- 
tain axioms  (§  22),  it  is  necessary  to  consider  the  precise 
limitations  of  these  axioms  before  proceeding  further. 

183.  Two  equations  are  said  to  be  equivalent  when  all  of 
the  roots  of  either  are  roots  of  the  other. 

E.g.,  x  +  4=2x-5, 

and  x  +  5  =  2  (a;  —  2),  are  equivalent  equations, 

for  x  =  9  is  a  root  and  the  only  root  of  each. 

But  X  =  2  and  x^  =  4  are  not  equivalent  equations,  for  —  2  is  a  root 
of  x2  =  4,  but  not  of  X  =  2. 

The  necessity  for  a  consideration  of  the  limitations  of 
the  use  of  the  axioms  is  seen  from  the  following : 
Suppose  1.  X  =  2. 

Then       2.  x2  =  4,  by  ax.  8. 

But  a  root  of  equation  2  is  not  necessarily  a  root  of  equation  1, 
because  while  equation  2  is  true  when  equation  1  is  true.,  it  is  not 
equivalent  to  equation  1. 

184.  Axioms  6  and  7.  If  equals  are  multiplied  or  divided 
hy  equals,  the  results  are  equal. 

This  is  true,  but  it  Tnust  not  he  interpreted  to  mean  that 
if  the  two  members  of  an  equation  are  multiplied  hy  equals, 
the  resulting  equation  is  equivalent  to  the  given  one. 

E.g.,  if  the  two  members  of  the  equation 
x-1  =  6 
are  multiplied  by  x  —  2,  we  have 

(X  -  1)  (X  -  2)  =  5  (X  -  2), 
or  x2-8x  +  12=0, 

or  (x  -  6)  (x  -  2)  =  0, 

which  has  two  roots,  x  =  2  and  x  =  6.  Of  these,  only  x  =  6  satisfies 
the  original  equation.  Hence,  the  resulting  equation  is  not  equivalent 
to  the  original  one ;  there  has  been  a  new  root  introduced. 


150  ELEMENTS  OF  ALGEBRA. 

185.  A  new  root  which  appears  in  performing  the  same 
operation  upon  both  members  of  an  equation  is  called  an 
extraneous  root. 

EXERCISES.    LXXVII. 

What,  if  any,  extraneous  roots  are  introduced  by  multi- 
plying both  members  of  the  following  equations  as  indi- 
cated ? 

1.  cc  +  2  =  5  hj   x-\-3. 

2.  X  —  a  =  0  ''     X  -\-  a. 

3.  a;2  -  1  =  0  "     x^-5x-\-6. 

4.  cc-2  =  4ic  +  l         "     3. 

5.  x-5  =  5x- 21       "     X. 

6.  x^-\-x  =(1  —  xy       "     x. 

7.  3x-4.  =  4:X-3      "     21. 

8.  (x  -f  ay  =  (x-  ay     "     fc -^  1. 

186.  Just  as  an  extraneous  root  may  be  introduced  by 
multiplying  both  members  of  an  equation  by  equals,  so  a 
root  may  also  be  lost  by  the  same  process. 

E.g.,  it  is  not  permissible  to  multiply  the  two  members  of  the 

equation 

(X  -  6)  (X  -  2)  =  0 

by  ,  expecting  thereby  to  obtain  an  equivalent  equation,  for  we 

X  —  Ji 

should  have  x  —  6  =  0, 

which  has  only  a  single  root,  x  =  6,  whereas  the  original  equation  had 
two  roots,  X  =  6  and  x  =  2.  Hence,  the  resulting  equation  is  not 
equivalent  to  the  original  one ;  a  root  has  been  lost  by  multiplying 
equals  by  equals. 

In  the  same  way,  while  if  we  multiply  both  members  of  the  equation 
x3  -  X  =  0 

by  - ,  ,  ,  or  ,  the  results  will  be  equal,  it  is  not  true 

X    x+1    X  —  1  x2  —  1 

that  we  shall  obtain  equivalent  equations. 


SIMPLE   EQUATIONS.  151 

187.  Hence,  it  appears  that  multiplying  the  two  members 
of  an  equation  f  (x)  =  0  by  a  function  of  x  does  not,  in  general, 
give  an  equivalent  equation.  The  operation  may  introduce 
an  extraneous  root,  or  it  may  suppress  a  root. 

It  should  also  be  stated,  in  connection  with  extraneous 
roots,  that  no  value  is  considered  a  root  unless  it  makes  the 
members  identically  equal.  Hence,  a  value  that  makes 
both  members  ififinite  is  not  a  root,  for  infinity  is  not  iden- 
tically equal  to  infinity  (§  166). 

2  X 

E.g.,  1  is  not  a  root  of  = ,  for  it  makes  each  member 

.^.^  x-lx  —  1 

mfmite. 

EXERCISES.    LXXVIII. 

Would  each  resulting  equation  be  equivalent  to  the  given 
one,  by  multiplying  both  members  as  indicated  below  ? 


1. 

^4 

X 

by 

X 

i' 

2. 

'iel^' 

x-2. 

3. 

hl*h' 

X. 

4. 

X^                  1 

X-1. 

x-1      x-1 

5. 

x^_6x  +  5 

x^-5x  +  4. 

5 

x^-5x  +  4.. 

6. 

x^-5x  +  6  = 

0 

1/(03-2). 

7. 

2x^-5  =  x^- 

-1 

1/(^  +  2). 

8. 

^2_3^_28. 

=  0 

1/(^-7). 

Q 

x'^  +  9x  +  14. 

1 

/^2     1      1J.  ^     t 

cc'^  +  14  ic  +  49      X  +  7        ^ 
jlO.    Also  by  cc^  +  14  a;  +  49  ;  also  by  a;  +  7. 


162  ELEMENTS   OF   ALGEBRA. 

188.  Axioms  8  and  9.  Like  powers  or  like  roots  of  equals 
are  equal. 

This  is  true,  but  it  must  not  be  interpreted  to  mean  that 
the  equation  formed  by  taking  like  powers  or  like  roots  of  the 
members  of  a  given  equation  is  equivalent  to  that  equation. 

E.g.,\t  x=l, 

then  x2  =  1 ,  or  x2  -  1  =  0,  or  (x  +  1)  (x  -  1)  =  0  ; 

but  this  equation  has  two  roots,  x  =  —  1,  and  x  =  +  1,  and  of  these, 
only  X  =  +  1  satisfies  the  original  equation. 

Similarly,  if  x^  =  4, 

it  is  true  that  x  =  2  ; 

but  this  equation  is  not  equivalent  to  the  original  one.     It  should  be 
written  x  =  +  2,  and  —  2. 

Students  are  liable  to  make  a  mistake  by  omitting  the 
±  sign  in  extracting  a  square  root,  thus  losing  a  root. 
E.g.,  in  the  equation 

x2  +  2  X  +  1  =  4, 
extracting  the  square  root,  x  +  1  =  2, 
.-.  X  =  1. 
It  should  be  x  +  1  =  +  2  and  -  2, 

.'.  X  =  1  and  —  3. 

EXERCISES.    LXXIX. 

What  extraneous  roots  are  introduced  by  squaring  both 
members  of  the  following  equations  ? 

1.    a;  =  0.  2.    ic  +  3  =  3.  3.    £c  -  2  =  2. 

4.    2 a;  =  9.  5.    a; -5  =  0.  6.    4.x  =-28. 

7.    205  +  1=3.        8.    ^  =  1.  9.    1  +  1  =  2. 

The  discussion  already  given  may  be  set  forth  in  four 
theorems.  These  theorems,  with  strict  proofs,  may  be 
found  too  abstract  for  most  beginners,  and  hence  they  are 
given  in  Appendix  V. 


SIMPLE   EQUATIONS.  '  153 

II.     SIMPLE  INTEGRAL  EQUATIONS. 

189.  General  directions  for  solution.  From  the  sugges- 
tions in  exs.  1-4,  on  p.  148,  it  appears  that,  to  solve  a 
simple  integral  equation,  we 

1.  Transpose  the  terms  containing  the  unknown  quan- 
tities to  the  first  member,  changing  the  signs  (axs.  2,  3, 
§§  22,  181)  ; 

2.  Transpose  the  terms  containing  only  known  quantities 
to  the  second  meiriber,  changing  the  signs  ; 

3.  Unite  terms  ; 

4.  Divide  by  the  coefficient  of  the  unknown  quantity  ; 

5.  Check  the  result  by  substituting  in  the  original  equation. 


EXERCISES.    LXXX. 

Solve  the  following  equations  : 

1.    «,a7  +  &  =  te  +  <^-  2.    {x  —  V)i\  —  X)  =■  —  x^ 

3.    8a; -(7 -a;)  =  29.  4.    3cc  -  2(2  -  a^)  =  21. 

5.    (2-x){5-x)  =  x\  6.    3(a;-l)  =  4(a;  +  l). 

7.  9x-3(5a;-6)  =  -30. 

8.  2(aj  +  3)-3(x  +  2)  =  0. 

9.  x{x'^  +  1)  =x (x^  —  1)  +  9. 

10.  {x  +  5)2  =:  21  a;  +  (4  -  xy\ 

11.  3a;  +  14.-5(a;-3)  =  4(a;  +  3). 

12.  a;(a;-l)-a;(aj-2)=2(a;-3). 

13.  a;(l +a;  +  a;2)  =  a;3_^a:2  +  3a;-17.5. 

14.  {x  +  1)  (a;  +  2)  =  (a;  +  3)  (a;  +  4)  -  50. 

15.  2(a;  +  l)  +  3(aj  +  2)  +  4(a;  +  3)=101. 

16.  {x  -  2)2  -{x-  3)2  =  (x-  4)2  -{x-  6)2. 


I 


154  ELEMENTS   OF  ALGEBRA. 

III.     SIMPLE  FRACTIONAL  EQUATIONS. 

190.  If  the  equation  contains  fractions,  these  may  be 
removed  by  multiplying  both  members  by  the  lowest  com- 
mon multiple  of  the  denominators.  This  is  called  clearing 
the  equation  of  fractions. 

Unless  the  fractions  are  in  their  lowest  terms  before  multiplying  by 
the  lowest  common  denominator,  an  extraneous  root  is  liable  to  be 
introduced  (§  187). 

It  is  not  always  advisable,  however,  to  clear  the  equation  of  frac- 
tions at  once,  as  is  seen  in  the  following  illustrative  problems. 

Illustrative  problems.      1.  An  equation  which  should  be 

^ 3      ic  -h  7 

cleared  of  fractions  at  once :    -^r-r — I — 7-=—  =  8. 

lo  IT 

1.  The  l.c.d.  of  the  fractions  is  15  •  17. 

2.  Multiplying  both  members  by  15-17,  by  ax.  6, 

17  a; -61  -|- 15  a -l- 105  =  15- 17 -8. 

3.  Subtracting  —51-1-  105,  and  uniting  terms, 

32aj=  1986. 

4.  Dividing  by  32, 

0^  =  62,1,. 

Check.    ^  +  ??^  =  8,  or  3H  +  4^^  =  8. 
15  17 

2.  An  equation  which  need  not  be  cleared  of  fractions  at 

X      3      1 
once:   x----  =  -' 

1.  Adding  f  and  uniting  terms, 

|x  =  f. 

2.  Multiplying  both  members  by  f  (or  dividing  both  members  by  f), 

«  =  ¥• 
Check.     1^-1  _|  =  |. 


7X-12 

8. 

Transposing  and  uniting  terms, 

7x- 

-4) 
12 

I 

Dividing  by  2  and  multiplying  by  7 

x-12 

I 

14  X 

-24  =  16x- 

30. 

4. 

f 

Adding  24- 15  X, 

-  X  =  -  6. 

h 

Multiplying  by  -  1, 

x  =  Q. 

Check.    H-ir%  =  l. 

SIMPLE  EQUATIONS.  155 

3.  An  equation  which  should  be  cleared  of  fractions  part 

^       .          3a;  +  7       2x-4:       x  +  1 
^.tatime:    -^^ ___  =  __. 

1.  Multiplying  both  members  by  15, 

Z.  +  7-'^^^^  =  Bz  +  S.  AX.0 


Azs.  2,  3 

Axs.  6,  7 
Ax.  2 
Ax.  6 


4.  An  equation  in  which  the  fractions  may  be  united  to 

,       ,        ,    ;        ,      .  X         x-\-l      x  —  S      x  —  9 

[vantage  before  clearing :  - — -  -  - — j  =  - — -  -  - — -  • 
X  —  J      x  —  1      x  —  o      x  —  7 

1.   Adding  the  fractions  in  each  member  separately, 

xa-x-x2+x  +  2      x«-15x  +  56-x«+ 15X-54 


I  It  will  be  noticed,  in  step  1,  that  the  bar  of  a  fraction  is  a  symbol  of 

K'egation,  and  in  adding  fractions  or  in  clearing  of  fractions  this 
t  be  tiiken  into  account. 
■— 


(x-2)(x-l)                         (x-6)(x- 

-7) 

5     .                             2             _             2 

(x-2)(x-l)      (x-6)(x-7) 

8.   Dividing  by  2  and  cleai'ing  of  fractions, 

xa_l3x  +  42=x2-3x  +  2. 

4.   .-.                               -10x=  -40. 

(Why  ?) 

6.    .-.                                       x=4. 

(Why  ?) 

4      5      -4       -5 

Check. = 

2      3-2-3 

156  ELEMENTS  OF  ALGEBRA. 

5.  An  equation  in  which  the  fractions  should  be  reduced 
to  mixed  expressions  before  clearing  : 

5a;-8      6x-U  _x-S      10.^-8 


1.    Reducing  to  mixed  expressions, 

2  2  2  2 

5  +  -^  +  6 =  1  -  ^—  +  10  + 


X  —  2  X  —  7  X  —  6  X  —  1 

2.    Subtracting  11  and  dividing  by  2, 


3.   Adding  the  fractions  in  each  member  separately, 
-5  -6 


{X  -  2)  (X  -  7)       (X  -  6)  (X  -  1) 

4.  Dividing  by  —  5,  and  clearing, 

x2-7x  +  6  =x2-  9x  +  14. 

5.  .-.  2  X  =  8.  /  (Why  ?) 

6.  .-.  X  =  4.  (Why  ?) 

lO  Of)  A  QO 

Check.     :L +  __  =  __  +  _,  or  6 +  6f  =  2  + lOf. 
EXERCISES.    LXXXI. 

First  determine  which  seems  the  best  method  of  solving 
each  of  the  following  equations ;  then  solve  and  (except  as 
the  teacher  otherwise  directs)  check  the  solution  by  substi- 
tuting the  roots  in  the  original  equation. 


1. 

ad 

X  —  1       X  —  b 

ax  +  bx 

-    a;  +  l       x-3 

3. 

50       12      49 
Ax^  X        lO' 

4.      2^1  =  2.4-15. 
X  —  o 

5. 

2  +  3-^^      4 

6.    0.5a:4-0.25rr  =  1.5. 

SIMPLE    EQUATIOJSS. 
X  .    1 


157 


^  +  ^  =  17-^ 
5^8         '       10 

4ic      5x 


9.    ^r- 


10. 


11. 


12. 


13. 


14. 


X  —  a  _  (2  ic  —  ay 
x-b  ~  {2x-  by ' 

1  +  i^       i^  +  1 


^»iC 

a 

iC 

^«^ 

3b-x 

0 

X 

+a ' 

2b  +  x 

6 

x  +  7 
12 

3x- 
4.x- 

4 
3 

+ 

X 

2* 

iC 

-2 

X-4: 

X 

6 

15. 


X        1  X 


16.    1 


17. 


18. 


19. 


a  -{-  X  -{- 


X' 


a  —  X 
3  4 


ic  —  3 


X  —  9      cc  —  6 
b^  —  aic 


5  +  a;Z'  —  ic_ 

&4-(x      b  ~  a       a^  —  b^ 

5  a;  4- 10.5  2x 


x  +  0.5         2ic  +  l 


=  9. 


158  ELEMENTS   OF   ALGEBRA. 


20. 


21. 


22. 


* 

2 
"3" 

2 
'3 

l^'  +  l 

%+x 

f  +  ^ 

X       ^x 

2 

+  4 
3 

=  7 

^x-2 
3 

8 

9 

15 

a;  +  3 

2x 

+  6 

ra^  +  2 

lx  +  5 

5 

X  — 

6      8-5a; 

23. 

6  4  12 

op;     2a;-5   ,   6x  +  3       ^  35 

25.   _^  +  ___  =  5x--. 

2a;  +  3       6a;  +  22  _  3a;  +  17 
5  15       ~5(l-a;)" 

a  —  c>  X  a  -\-h  x 

28     ^  +  46^  +  ^*       4a7  +  a  +  2^>  ^ ' 
a;  +  a  +  ^  iz;-[_(t  —  ^     ""* 

29.        1  2  3  4 


-1      x-2      x-3  x-4. 

30     '^  +  3       a;-6_g;  +  4  a; -5 

a;  +  l      a;-4      a;  +  2  a;  -  3 

1  1 


31. 


(a;  H-  3)  (a;  +  5)       (a;  +  9)  (a;  -  5) 


32     1_+  ^^      9  -  11  a;  ^  14(2 a;  -  3)^ 
5  + 7a;        5  -  7  a;  ""    25  -  49  a;^  ' 

33.    -^ 1 l  +  _^_  =  o 

2a; -1      a;-3      a;2a;-5 


SIMPLE   EQUATIONS. 


159 


34 
35 
36. 

37. 

38. 

(9. 

40. 

1. 

42. 
43. 
44. 

45. 


5  cc h  1  =  3  a;  H ^ V-  7. 


5  4       ^1/    3  8     \ 

Bx  +  S      2ic  +  3~5Vic  +  3      a;  +  2/ 

,       a;/,        3\      6xf^        6\      „, 
^-2(l-4^j  =  Ti^-7^h''^^ 
3a;'^-2a;  +  l_  (7a;  -  2)  (3a;  -  6)       _9 


35 


+ 


10 


a{a  +  b)x       cu'-lP'        2bx   _{5a  +  b)b 
a^  —  b'^  a  +  b        b  —  a  a  —  b 

\x-l^-^x-l  =  ^%  +  \^x-\l-^x. 

13.r-10      4a; +  9       7  (x  -  2)  _  13a;  -  28 
36  18  12       ~  17a; -66* 

a  2(3a  +  5)       8^4- 15  _  3(^  +  2)  1 

a;  +  1  a;  —  1  a;  —  2     ~     a;  —  3         a;  +  2* 

TV(2^-l)-TV(3^-2)  =  TV(^-12)-,,\(a;  +  l). 

(g  +  bfjx  +  1)  +  (g  +  ^,)(a;  +  1)  +  (^  +  1) 
a  +  b  +  1 

=  («  4-  by  +  (g  +  &)  +  1. 


a;  H-1  _lf        a;  +  1 

3    ~2y~~Y~ 


x-2 


\{'-'-^) 


+ 


31 


46. 


5  3V^         2      y   ■   60 

3  a;5  +  12  a;*  +  44  a;^  +  185  a;^  +  8  a;  +  98 


I 


3  a;*  +  18  a;«  +  26  a;2  +  15  a;  +  14 


3a;^  +  44a;  +  2 
3a;2  +  6a;  +  2 


160 


ELEMENTS   OF  ALGEBRA. 


IV.     IRRATIONAL  EQUATIONS    SOLVED   LIKE  SIMPLE 
EQUATIONS. 

191.  It  often  happens  that  irrational  equations  can  be 
reduced  to  equivalent  simple  equations  and  thus  solved. 

E.g. ,  Vx  =  2  can  be  reduced  to  the  equivalent  simple  equation  a  =  4. 
In  applying  ax.  8  it  is  possible,  however,  that  extraneous  roots  may  be 
introduced  (§  185).  That  this  is  not  the  case  in  this  instance  is  seen 
by  substituting  the  value  of  x  in  the  original  equation. 

192.  A  question  at  once  arises,  however,  in  dealing  with" 
equations  like 


Vic'^  +  2  ic  +  1  +  Vic^  _  2  ic  +  1  =  4. 
Shall  this  be  reduced  to 

or  shall  only  the  positive  roots  be  considered,  as  in 

iC  +  l+£C  —  1=4? 

The  former  would  give  x  =  ±2,  the  latter  only  x  =  2. 

To  answer  this  question,  let   ^f(x)   and    -\/F{x),  foi 
brevity,  represent  the  square  roots  of  any  two  fimctionsi 
of  x,  like  those  already  mentioned. 

Then  it  is  evident  that  an  irrational  equation  of  the  foi 


-s/f(x)  +  VF(x)  = 
involves  four  equations,  viz.: 
1. 
2. 
3. 
4.  -■\/f(xj--^F(x) 

where  V/(£c)  and  ^F{x)  represent,  in  these  four  equations, 
only  the  positive  square  roots. 

This  is  also  seen  in  the  case  of  Vi  +  Vo,  which  equals  (±  2)  +  (±  3).| 


SIMPLE   EQUATIONS.  161 

m 

"•     193.    Hence,  any  root  which  satisfies  any  one  of  the  four 

equations  is  strictly  a  root  of  V/(a;)  +  ■y/F(x)  =  a. 

By  convention,  however,  only  the  roots  which  satisfy  equa- 
tion 1  are  usually  considered. 


For  example,  consider  the  equation  ■\x  —  2-{-  Va;  —  5  =  1. 


1.  Vx  -  5  =  1  -  Vx  -  2.  Ax.  3 


2.  .-.  x-5  =  l+x-2-  2  Vx  -  2.  Ax.  8 

3.  .-.  2Vx  -2  =  4.  Ax.  3 

4.  .-.  X  -  2  =  4.  Axs.  7,  8 

5.  .-.  x  =  Q>. 
Substituting  6  for  x  in  the  given  equation, 

Vi  +  Vl  =  1,  or  (db  2)  +  (±  1)  =  1. 

While  this  is  true  in  the  form  (+  2)  +  (—  1)  =  1,  the  root  6,  by  the 
convention  just  given,  is  usually  called  extraneous. 

194.  Irrational  equations  can  often  be  solved  by  isolating 
the  radical  and  then  applying  ax.  8.  For  example,  consider 
the  equation  Vic  —  2  —  Vcc  —  5  =  1. 

1.  We  first  isolate  the  radical  Vx  —  2,  by  adding  Vx  —  5  to  each 
member. 


2.  .-.  Vx  -  2  =  1  +  Vx  -  5. 

3.  Then,  by  squaring  both  members, 
x-2  =  l+x-5  +  2Va 


l.   Then,  isolating  the  radical  Vx  —  5,  by  subtracting  1  +  x  —  5 

dividing  by  2,  

1  =  Vx-6. 

5.    .•.  \  z=  X  —  5,  whence  x  =  6. 


Check.     V6-2  _  Ve  -  5  =  1. 

L95.    If  the  equation  contains  several  irrational  expres- 
is,  there  is  no  general  rule  for  solution.     The  student 
ist  use  his  judgment  as  to  which  radical  it  is  best  to 
llate  first. 


162  ELEMENTS  OF  ALGEBRA. 

Illustrative  problems.     1.  Solve  the  equation 


Va;  +  1  —  4  Vcc  —  4  +  5  Vic  —  7  =  0. 


1.  Isolating  the  radical  4  vcc  —  4  "by  adding  it  to  both  members,  we 
have :  

Vx  +  l  +  5Vx-7  =  4  Vx  -4. 

2.  Squaring 

X  +  1  +  25x  -  175  +  10  Vx2-6x-7  =  16x  -  64. 

3.  .-.  X  -  11  =  -  Vx2  -  6  X  -  7.  (Why  ?) 
4..-.  x2-22x+ 121  =x2-6x-7.  (Why?) 
5.  .-.  x  =  8.  (Why?) 
Check.  V9-4V4  +  5V1  =3-8  +  5  =  0. 

2.  Solve  the  equation  Vcc  =  —  2. 

Squaring,  x  =  4.  But  on  substituting  4  for  x,  Vi  =  —  2.  This 
satisfies  the  equation  because  Vx  is  both  +  2  and  —  2.  But  since  the 
positive  sign  is  usually  taken  with  the  radical  (§  193),  4  is  usually 
called  an  extraneous  root  and  the  equation  is  said  to  be  impossible. 
The  equation  —  Vx  =  —  2  is  not  open  to  the  same  objection  for  it  is 
satisfied  by  x  =  4. 

EXERCISES.    LXXXII. 

Solve  the  following  equations,  designating  such  roots  as 
are  usually  called  extraneous. 


1-.    Vic  +  2  -  Va;  +  9  =  7. 
2.    -y/x  +  Va  -\-  X  =  a/  V^. 


3.    Vx  +  19  +  Vcc  +  S 


4.    2  Vcc  -  1  +  V4ic  +  5  =  9. 


5.  V8^+5-2  V2x-l  =  l.* 

6.  4  ^x  +  2  -  Vx  +  7  -  5  Vic-1. 


SIMPLE   EQUATIONS.  163 

V.    APPLICATION  OF  SIMPLE  EQUATIONS. 

A.     Problems  Eelating  to  Numbers. 

Illustrative  problems,     1.  The  sum  of  two  numbers  is  200, 
and  their  difference  is  50.     Find  the  numbers. 


1. 

Let 

X  =  the  lesser  number. 

2. 

Then 

X  +  50  =  the  greater  number. 

3. 

And 

X  +  X  +  50  =  the  sum. 

4. 

But 

200  =  the  sum. 

5. 

X  +  X  +  50  =  200.                   -  , 

6. 

.-. 

x  =  75, 

and 

X  +  50  =  125. 

Check.     The  sum  of  125  and  75  is  200,  and  their  difference  is  50. 

Always  clieck  by  substituting  in  the  problem  instead  of 
the  equation,  because  there  may  have  been  an  error  in 
forming  the  equation.  The  neglect  to  take  this  precaution 
often  leads  to  wrong  results. 

2.  What  number  must  be  added  to  the  two  terms  of 
the  fraction  ^^  in  order  that  the  resulting  fraction  shall 
equal  |f  ? 

1.  Let  X  =  the  number  to  be  added. 

2.  Then  1+x  ^  59_ 

23  +  X      67 

3.  .-.  67  (7  +  X)  ==  59  (23  +  x).  (Why  ?) 

4.  .-.  469  +  67x=  1357  + 59  X. 

5.  .-.  8x  =  888.  (Why?) 

6.  .-.  x  =  lll.  (Why?) 

Check.      ^"^^^^   ^  H?  ^  ^.     That  Is,  if  111  is  added  to  both 
23  +  111       134      67 
terms  of  the  fraction  2^3 ,  the  result  equals  ff .  * 


164  ELEMENTS   OF  ALGEBRA. 

EXERCISES.    LXXXIII. 

1.  What  number  is  that  which  when  subtracted  from 
28  gives  the  same  result  as  when  divided  by  28  ? 

2.  Or,  more  generally,  what  number  is  that  which  when 
subtracted  from  n  gives  the  same  result  as  when  divided 
by  n  ?     Check  by  supposing  that  n  =  S,  n  =  28. 

3.  What  number  is  that  which  when  multiplied  by  16 
gives  the  same  result  as  when  added  to  16  ? 

4.  Or,  more  generally,  what  number  is  that  which  when 
multiplied  by  7i  gives  the  same  result  as  when  added  to 
n  ?     Check  by  supposing  that  n  =  2,  n  =  16. 

5.  What  number  is  that  which  when  divided  by  12 
gives  the  same  result  as  when  added  to  12  ? 

6.  Generalize  ex.  5  and  check.     (See  exs.  2,  4.) 

7.  What  number  is  that  which  when  subtracted  from 
25  gives  the  same  result  as  when  multiplied  by  25  ? 

8.  Generalize  ex.  7  and  check.     (See  exs.  2,  4,  6.) 

9.  What  number  must  be  added  to  3  and  7  so  that  the 
first  sum  shall  be  f  of  the  second  ? 

10.  Or,  more  generally,  what  number  must  be  added  to 

a  and  b  so  that  the  first  sum  shall  be  —  of  the  second  ? 

n 

Check  by  supposing  that  a  =  3,  b  =  7 ,  77i  =  3,  n  =  4:. 

11.  Determine  x,  knowing  that  a^  —  5a^-{-4:a  —  x  is 
algebraically  divisible  by  2  a-  -|-  1. 

12.  Divide  the  number  121  into  two  parts  such  that  the 
greater  exceeds  the  less  by  73. 

13.  Or,  more  generally,  divide  the  number  n  into  two 
parts  such  that  the  greater  exceeds  the  less  by  a. 


SIMPLE   EQUATIONS.  165 

14.  Divide  the  number  121  into  three  parts  such  that 
the  first  exceeds  the  second  by  85  and  the  second  is  four 
times  the  third. 

15.  Divide  the  number  n  into  three  parts  such  that  the 
first  exceeds  the  second  by^  and  the  third  by  q.  Check  by 
letting  7i  =  10,  i?  =  1,  q  =  1. 

16.  What   is  the  value  of   n  if  ;- — — -  =  — -  when 

17.  If  each  of  the  two  indicated  factors  of  the  two 
unequal  products  52-45  and  66-37  is  diminished  by  a  cer- 
tain number,  the  two  products  are  equal.  What  is  the 
number  ? 

\  18.  Divide  the  number  99  into  four  parts  such  that  if 
2  is  added  to  the  first,  subtracted  from  the  second,  and 
multiplied  by  the  third,  and  if  the  fourth  is  divided  by  2, 
the  results  shall  all  be  equal. 

tt .  19.  Or,  more  generally,  divide  the  number  n  into  four 
parts  such  that  if  a  is  added  to  the  first,  subtracted  from 
the  second,  and  multiplied  by  the  third,  and  if  the  fourth 
is  divided  by  a,  the  results  shall  all  be  equal.  Check  by 
letting  n  =  10,  a  =  1. 

20.  The  square  of  a  certain  number  is  1188  larger  than 
that  of  6  less  than  the  number.     What  is  the  number  ? 

21.  The.  square  of  13  times  a  certain  number,  less  the 
square  of  3  more  than  12  times  the  number,  equals  the 
square  of  9  less  than  5  times  the  number.  What  is  the 
number  ? 

22.  What  number  must  be  added  to  each  term  of  the 
fraction  y  that  it  may  equal  the  fraction  -?  Check  by 
letfmg  a  =z  3,  h  =  5,  c  =  ^,  d  =  10. 


166  ELEMENTS   OF  ALGEBRA. 

B.     Problems  Relating  to  Common  Life. 

Illustrative  problems.  1.  What  sum  gaining  6|^%  of  itself 
in  a  year  amounts  to  $157.50  in  2  yrs.  ? 

1.  Let  a;  =  the  wwm&er  of  dollars. 

2.  Then  6J%  x  =  the  number  of  dollars  of  interest  for  1  yr. 

3.  .-.      x  +  2-6i%x  =  157.50.  (Why?) 

4.  .-.  1.12^x^157.50. 

5.  .-.  X  =  140.  (Why  ?) 
Check.     The  interest  on  $140  for  2  yrs.  at  6^%  is  §17.50,  and  hence 

the  amount  is  $157.50. 

2.  The  cost  of  an  article  is  $17.15,  and  this  is  30%  less 
than  the  marked  price.     What  is  the  marked  price  ? 

1.  Let  X  =  the  number  of  dollars  of  marked  price. 

2.  Then  30%  x  =  the  number  of  dollars  of  discount. 

3.  .-.  x-30%x  =  17.15. 

•4.    .-.  0.7x  =  17.15.  (Why?) 

5.    .-.  x  =  24.50.  (Why?) 

Check.     $24.50  less  30%  of  $24.50  is  $17.15. 

EXERCISES.    LXXXIV. 

23.  What  is  that  sum  which  diminished  by  9|-%  of  itself 
equals  $1538.50  ? 

24.  How  long  will  it  take  an  investment  of  $6024  to 
amount  to  $7658.01  at  3|-%   simple  interest? 

25.  A  man  invests  f  of  his  capital  at  4%  and  the  rest 
at  3^%,  and  thus  receives  an  annual  income  of  $76.  What 
is  his  capital  ? 

26.  A  man  invests  one-fourth  of  his  capital  at  5<^,  one- 
fifth  at  4%,  and  the  rest  at  3%,  and  thus  secures  an  animal 
income  of  $3700.     What  is  his  capital  ? 


SIMPLE   EQUATIONS.  167 

*'  27.  A  train  traveling  30  mi.  per  hour  takes  2|  hrs.  longer 
to  go  from  Detroit  to  Chicago  than  one  which  goes  -J  faster. 
What  is  the  distance  from  Detroit  to  Chicago  ? 

28.  A  loaned  to  B  a  certain  sum  at  4-|-^,  and  to  C  a  suii^ 
$200  greater  at  5%  ;  from  the  two  together  he  received 
$276  per  annum  interest.     How  much  did  he  lend  each  ? 

29.  The  interest  for  8  yrs.  upon  a  certain  principal  is 
$1914,  the  rate  being  3|-%  for  the  first  year,  3^%  for  the 
second,  3f  %  for  the  third,  and  so  on,  increasing  :^^  each 
year.     What  is  the  principal  ? 

30.  A  bicyclist  traveling  a  miles  per  hour  is  followed, 
after  a  start  of  m  mi.,  by  a  second  bicyclist  traveling  h  mi. 
per  hour,  h  >  a.  At  these  rates,  in  how  many  hours  after 
the  second  starts  will  he  overtake  the  first  ? 

31.  A  capitalist  has  f  of  his  money  invested  in  mining 
stocks  which  pay  him  13%,  ^  in  manufacturing  which  pays 
him  9%,  and  the  balance  in  city  bonds  which  pay  him  3%. 
What  is  his  capital,  if  his  total  income  is  $26,640  ? 

32.  A  man  spends  -th  of  his  income  for  food,  -th  for 

a  0 

rent,  -th  for  clothing,  -th  for  furniture,  and  saves  e  dollars. 

c  (t 

How  much  is  his  income  ? 

33.  Two  trains  start  at  the  same  time  from  Buffalo  and 
New  York,  respectively,  450  mi.  apart ;  the  one  from  New 
York  travels  at  the  rate  of  50  mi.  per  hour,  and  the  other 
0.8  as  fast.     How  far  from  New  York  do  they  meet  ? 

34.  Two  trains  start  at  the  same  time  from  Syracuse, 
one  going  east  at  the  rate  of  35  nii.  per  hour  and  the 
other  going  west  at  a  rate  \  greater.  How  long  after  start- 
ing will  they,  at  these  rates,  be  exactly  100  mi.  apart  ? 


168  ELEMENTS  OF  ALGEBRA. 

35.  A  train  runs  75  mi.  in  a  certain  time.  If  it  were  to 
run  2^  mi.  an  hour  faster,  it  would  run  5  mi.  farther  in  the 
same  time.     What  is  the  rate  of  the  train  ? 

36.  A  steamer  can  run  25  mi.  an  hour  in  still  water.  If 
it  can  run  90  mi.  with  the  current  in  the  same  time  that  it 
can  run  60  mi.  against  the  current,  what  is  the  rate  of  the 
current  ? 

37.  The  cost  of  publication  of  each  copy  of  a  certain 
illustrated  magazine  is  6^  cts. ;  it  sells  to  dealers  for  6  cts., 
and  the  amount  received  for  advertisements  is  10%  of  the 
amount  received  for  all  magazines  issued  beyond  10,000. 
Find  the  least  number  of  magazines  which  can  be  issued 
without  loss. 

38.  A  steamer  and  a  sailboat  go  from  M  to  N,  the 
former  at  the  rate  of  35  mi.  in  3  hrs.  and  the  latter  at  the 
rate  of  10  mi.  in  the  same  time.  The  sailboat  has  a  start 
of  3^  mi.,  but  arrives  at  N  5  hrs.  after  the  steamer.  How 
long  did  it  take  the  steamer  to  go  from  M  to  N,  and  what 
is  the  distance  ? 

39.  Two  engines  are  used  for  pumping  water  from  dif- 
ferent shafts  of  a  mine,  their  combined  horse  power  being 
represented  by  108.  The  first  engine  pumps  22  gals,  every 
10  sees,  from  a  depth  of  310  yds. ;  the  second  pumps  9  gals, 
more  in  the  same  length  of  time  from  a  depth  of  176  yds. 
Eequired  the  horse  power  of  each. 

40.  There  are  two  hoisting  engines  at  a  coal-pit  mouth, 
the  first  capable  of  raising  at  the  rate  of  144  tons  every  5 
hrs.  from  a  depth  of  375  ft.,  and  the  second  80  tons  every 
3  hrs.  from  a  depth  of  540  ft.  After  the  first  had  been 
running  If  hrs.  the  Second  began,  and  after  7  hrs.  it  had 
raised  from  the  bottom  of  the  mine  11:J-  tons  more  than  the 
first.     Eequired  the  depth  of  the  mine. 


SIMPLE   EQUATIONS.  169 

C.     Problems  Eelating  to  Science. 

Illustrative  problems.  1.  Alcohol  is  received  in  the  labo- 
ratory 0.95  pure.  How  much  water  must  be  added  to  a 
gallon  of  this  alcohol  so  that  the  mixture  shall  be  0.5  pure  ? 

1.  Let  X  =  the  number  of  gallons  of  water  to  be  added. 

2.  Then  0.5(1  +  x)  represents  the  alcohol  in  the  mixture. 

3.  But  0.95  represents  the  alcohol  in  the  original  gallon. 

4.  .-.  0.5(1 +  x)  =  0.95. 

5.  .-.  x  =  0.9.  (Why?) 
Check.     Adding  0.9  gal.,  there  are  1.9  gals,  of  the  mixture,  0.5  of 

which  is  the  0.95  gal.  of  alcohol. 

2.  Air  is  composed  of  21  volumes  of  oxygen  and  79  >ol- 
mnes  of  nitrogen.  If  the  oxygen  is  1.1026  times  as  heavy 
as  air,  the  nitrogen  is  what  part  as  heavy  as  air  ? 

1.  21 . 1.1026  +  79  X  =  100.  (Why  ?) 

2.  .-.  x  =  0.9727.  (Why?) 

EXERCISES.    LXXXV. 

41.  How  much  water  must  be  added  to  a  5%  solution  of 
a  certain  medicine  to  reduce  it  to  a  1  %  solution  ? 

42.  How  much  pure  alcohol  must  be  added  to  a  mixture 
of  f  alcohol  so  that  -^^  of  the  mixture  shall  be  pure  alcohol  ? 

43.  In  midwinter  in  St.  Petersburg  the  night  is  13  hrs. 
longer  than  the  day.  How  many  hours  of  day  ?  of  night  ? 
At  what  time  does  the  sun  rise  ?    set  ? 

44.  How  many  ounces  of  silver  700  fine  (700  parts  pure 
silver  in  1000  parts  of  metal)  and  how  many  ounces  900 
fine  must  be  melted  together  to  make  78  oz.  750  fine  ? 

45.  How  many  ounces  of  pure  silver  must  be  melted 
with  500  ozT  of  silver  750  fine  to  make  a  bar  900  fine  ? 


170  ELEMENTS  OF  ALGEBRA. 

46.  How  many  pounds  of  pure  water  must  be  added  to 
32  lbs.  of  sea  water  containing  16%  (by  weight)  of  salt,  in 
order  that  the  mixture  shall  contain  only  2  %  of  salt  ? 

47.  In  a  certain  composition  of  metal  weighing  37.5  lbs., 
18f  %  is  pure  silver.  How  many  pounds  of  copper  must 
be  melted  in  so  that  the  composition  shall  be  only  15.625% 
pure  silver  ? 

48.  How  many  pounds  of  copper  should  be  melted  in 
with  94.5  lbs.  of  an  alloy  consisting  of  3  lbs.  of  silver  to 
4  lbs.  of  copper  so  that  the  new  alloy  shall  consist  of  7  lbs. 
of  copper  to  2  lbs.  of  silver  ? 

49.  What  per  cent  of  the  water  must  be  evaporated  from 
a  6%  solution  of  salt  (salt  water  which  contains  6%,  by 
weight,  of  salt)  so  that  the  remain,ing  portion  of  the  mix- 
ture may  be  a  12%  solution  ? 

50.  The  planet  Venus  passes  about  the  sun  13  times  to 
the  earth's  8.  How  many  months  from  the  time  when 
Venus  is  between  the  earth  and  the  sun  to  the  next  time 
when  it  is  in  the  same  relative  position  ? 

51.  Two  bodies  start  at  the  same  time  from  two  points 
243  in.  apart,  and  move  towards  each  other,  one  at  the  rate 
of  5  in.  per  second,  and  the  other  2  in.  per  second  faster. 
In  how  many  seconds  will  they  be  39  in.  apart  ? 

52.  Prom  two  points  d  units  apart  two  bodies  move 
towards  each  other  at  the  rate  of  a  and  h  units  a  second, 
respectively.  After  how  many  seconds  will  they  be  c 
units  apart  for  the  first  time  (c  <  d)?  together  ?  c  units 
apart  for  the  second  time  ? 

53.  These  bodies  (of  ex.  52)  move,  from  the  two  starting 
points,  away  from  one  another.  How  far  are  they  apart 
after  t  sees.  ?     When  will  they  be  e  units  apart  (e>d)? 


t 


SIMPLE   EQUATIONS.  171 


54.  If  sound  travels  5450  ft.  in  5  sees,  when  the  temper- 
ature is  32°,  and  if  the  velocity  increases  1  ft.  per  second 
for  every  degree  that  the  temperature  rises  above  32°,  how- 
far  does  sound  travel  in  8  sees,  when  the  temperature  is 
70°? 

55.  Seen  from  the  earth,  the  moon  completes  the  circuit 
of  the  heavens  in  27  das.  7  hrs.  43  mins.  4.68  sees.,  and  the 
sun  in  3(35  das.  5  hrs.  48  mins.  47.8  sees.,  in  the  same  direc- 
tion. Required  the  time  from  one  full  moon  to  the  next, 
the  motions  being  supposed  to  be  uniform.  Answer  cor- 
rect to  0.0001  da. 

56.  In  Spitzbergen  (77°  N.  lat.)  there  is  a  certain  part  of 
the  year  in  which  the  sun  does  not  rise,  remaining  con- 
stantly below  the  horizon ;  there  is  also  an  equal  length  of 
time  during  which  it  does  not  set.  The  period  in  which  it 
rises  and  sets  is  1^  months  longer  than  the  period  of  con- 
tinued night.  How  many  months  in  each  of  these  three 
divisions  of  the  year  ? 

57.  It  is  shown  in  physics  that  if  ^  =  the  number  of 
seconds  which  it  takes  a  pendulum  to  swing  from  one  state 
of  rest  to  the  next,  through  a  small  angle,  then  t  =  7r  \l/g, 
where  7r=3^,  <7  =  32.2,  and  Z=the  number  of  feet  of  length 
of  the  pendulum.  Required  the  length  of  a  1-second  pen- 
dulum; of  a  2-seconds  pendulum;  of  a  pendulum  which 
oscillates  56  times  in  55  sees. 

58.  It  is  proved  in  physics  that  if  v  =  the  velocity  of  a 
body  which  started  with  an  initial  velocity  of  a  ft.  per 
second  and  has  gained  in  velocity  /  ft.  per  second  for  t 
seconds,  then  v  =  a  -{-  ft.  Suppose  v  =  15,  a  =  0,  t  =  5. 
Find  /.  (This  is  one  of  many  exceptions  to  the  custom  of 
representing  known  quantities  by  the  first  and  unknown 
(quantities  by  the  last  letters  of  the  alphabet.) 


172  ELEMENTS  OF  ALGEBRA. 

D.     Problems  Eelating  to  Mensuration. 

The  following  formulas  are  proved  true  in  geometry  and 
are  probably  already  known  to  the  student  from  his  work 
in  arithmetic.     They  are  inserted  for  reference. 

Symbols. 

TT  =  3.14159  . . .,  or  nearly  Sf 

r  =  radius.  a  =  area.  b  =  base. 

c  =  circumference.         h  =  altitude  (height). 

Formulas. 

Rectangle,  a  =  bh.  Triangle,  a  =  ^bh. 

The  square  on  the  hypotenuse  of  a  right-angled  triangle 
equals  the  sum  of  the  squares  on  the  other  two  sides. 
Circle,  c  =  2  irr.  a  =  rrrr^. 

Illustrative  problem.  What  is  the  length  of  the  radius  of 
the  circle  whose  circumference  is  62.8318  units  ? 

1.  •.•  0  =  27^*, 

2.  .-.  62.8318  =  2 -3. 14159- r. 

3.  .-.  10  =  r. 

EXERCISES.    LXXXVI. 

59.  What  is  the  altitude  of  a  triangle  whose  area  is 
7  sq.  in.  and  whose  base  is  2  in.  ? 

60.  What  is  the  length  of  the  base  of  a  rectangle  whose 
area  is  18  sq.  in.  and  whose  altitude  is  2^  in.  ? 

61.  From  the  top  of  a  flagstaff  a  line  just  reaches  the 
ground ;  if  a  line  a  yard  long  is  tied  to  this  (no  allowance 
being  made  for  the  knot),  the  whole  line  when  tightly 
stretched  touches  the  ground  20  ft.  from  the  staff.  Ee- 
quired  the  height  of  the  staff. 


I 


SIMPLE   EQUATIONS.  173 

62.  What  is  the  length  of  the  radius  of  the  circle  whose 
area  contains  25  tt  sq.  in.  ? 

63.  If  the  area  of  a  triangle  is  3  V3,  and  the  base  is 
2  V3,  required  the  altitude. 

64.  What  is  the  length  of  the  diameter  of  the  circle 
whose  circumference  is  157.0795  in.? 

65.  The  perimeter  of  a  rectangle  is  14  in.,  and  the  base 
is  33-^%  longer  than  the  altitude.  Eequired  the  length  of 
the  diagonal. 

66.  Two  rectangles  of  the  same  area  have  the  following 
dimensions ;  the  first,  15  ft.  by  10  ft.,  and  the  second, 
18  ft.  by  X  ft.     Eequired  x. 

67.  What  is  the  length  of  the  radius  of  the  circle  the 
number  of  square  units  of  whose  area  equals  the  number 
of  linear  units  of  circumference  ? 

68.  The  perimeter  of  a  triangle  is  75  in. ;  the  second 
side  is  §  of  the  first  and  the  third  is  f  of  the  first. 
Required  the  length  of  each  side. 

69.  The  area  of  a  triangle  is  250  sq.  ft.,  and  the  altitude 
is  25%  more  than  the  base.  Eequired  the  length  of  the 
base.     Is  the  resulting  equation  linear  ? 

70.  The  perimeter  of  a  triangle  is  24  in.,  the  first  side  is 
2  in.  longer  than  the  second,  and  the  second  is  2  in.  longer 
than  the  third.     Eequired  the  length  of  each  side. 

71.  A  dock  pile  is  |  above  water  and  ^  is  driven  into 
the  soil ;  if  the  water  at  the  dock  is  7  ft.  deep,  what  is  the 
entire  length  of  the  pile  and  how  many  feet  are  above 
water  ? 


174  ELEMENTS  OF  ALGEBRA. 

E.     Historical  Problems. 

Many  problems  which  were  of  considerable  difficulty 
prior  to  the  introduction  of  our  present  algebraic  symbols, 
about  the  opening  of  the  seventeenth  century,  are  now  com- 
paratively easy.  They  have  considerable  historical  interest 
as  showing  the  state  of  the  science  at  various  periods,  and 
a  few  examples  are  here  inserted. 

EXERCISES.    LXXXVII. 

72.  If  9  porters  drink  12  casks  of  wine  in  8  das.,  how 
many  will  last  24  porters  30  das.  ?  (Tartaglia,  a  famous 
Italian  algebraist,  about  1550  a.d.) 

73.  Demochares  lived  i  of  his  life  as  a  boy,  ^  as  a 
young  man,  ^  as  a  man,  and  13  years  as  an  old  man.  How 
old  was  he  then  ?     (Metrodorus,  325  a.d.) 

74.  Of  4  pipes,  the  first  fills  a  cistern  in  1  da.,  the  second 
in  2  das.,  the  third  in  3  das.,  and  the  fourth  in  4  das.  How 
long  will  it  take  all  running  together  to  fill  it  ? 

75.  In  the  center  of  a  pond  10  ft.  square  grew  a  reed 
1  ft.  above  the  surface ;  but  when  the  top  was  pulled  to 
the  bank  it  just  reached  the  edge  of  the  surface.  How 
deep  was  the  water  ?  (From  an  old  Chinese  arithmetic, 
Kiu  chang,  about  2600  b.c.) 

76.  A  horse  and  a  donkey,  laden  with  corn,  were  walk- 
ing together.  The  horse  said  to  the  donkey  :  "  If  you  gave 
me  one  measure  of  corn,  I  should  carry  twice  as  much  as 
you;  but  if  I  gave  you  one  we  should  carry  equal  bui-dens." 
Tell  me  their  burdens,  0  most  learned  master  of  geometry. 
(Attributed  to  Euclid,  the  great  writer  on  geometry  at 
Alexandria,  about  300  b.c.) 


I 

eq 

i 


SIMPLE   EQUATIONS.  175 

77.    Heap,  its  whole,  its  seventh,  it  makes  19.     (That  is, 

hat  is  the  number  which  when  increased  by  its  seventh 

equals  19?     From  the  mathematical  work  copied  by  the 

Igyptian  Ahmes  about  1700  b.c.  from  a  papyrus  written 

ut  a  thousand  years  earlier.) 


I 


78.  Find  the  number,  -^  of  which  and  1,  multiplied  by  ^ 
of  which  and  2,  equals  the  number  plus  13.  (Mohammed 
ben  Musa  Al-Khowarazmi,  the  famous  Persian  mathemati- 
cian, 800  A.D.  From  the  title  of  his  book  comes  the  word 
Algebra,  and  from  the  latter  part  of  his  name  —  referring 

his  birthplace  —  comes  our  word  Algorism.) 


79.  In  a  pond  the  top  of  a  lotus  bud  reached  ^  ft.  above 
e  surface,  but  blown  by  the  wind  it  just  reached  the 
surface  at  a  point  2  ft.  from  its  upright  position.  How 
deep  was  the  water?  (From  a  mathematical  work  by 
Bhaskara,  a  Hindu  writer  of  about  1150  a.d.     The  work 

Kas  named  the  Lilavati  in  honor  of  his  daughter.) 
80.    Two  anchorites  lived  at  the  top  of  a  perpendicular 
iff  of  height  h,  whose  base  was  mh  distant  from  a  certain 
•wn.     One  descended  the  cliff  and  walked  to  the  town; 
the  other  flew  up  a  height,  x,  and  then  flew  directly  to 

ibe  town.  The  distance  traversed  by  each  was  the  same. 
I^nd  X.  (Brahmagupta,  a  Hindu  mathematician,  about 
^0  A.D.) 
^  81.  An  ancient  problem  relates  that  Titus  and  Caius  sat 
iOwn  to  eat,  Caius  furnishing  7  portions  and  Titus  8,  all 
f  equal  value.  Before  they  began  Sempronius  entered 
and  they  all  ate  equally  and  finished  the  food.  Sempronius 
then  laid  down  30  denarii  (pence)  and  said  :  "  Divide  these 

K[uitably  between  you  in  payment  for  my  meal."     How 


176  ELEMENTS   OF  ALGEBRA. 

F.     Discussion  of  Problems. 

196.  Many  problems  can  be  suggested  which  admit  of 
mathematical  solution,  but  whose  practical  solutions  are 
impossible  by  reason  of  the  physical  conditions  imposed. 

E.g.^  I  can  look  out  of  the  window  18  distinct  times  in  4  sees. 
What  is  the  rate  per  second  ? 

The  answer,  4|-  times  per  second,  while  entirely  correct  from  the 
mathematical  standpoint,  is  physically  impossible ;  for  while  I  can  look 
out  4  times,  I  cannot  look  out  half  of  a  time. 

The  problem  might  easily  be  changed,  however,  so  as  to  demand 
the  time  required  to  look  out  once,  the  answer  behig  |  of  a  second. 

A  similar  absurdity  appears  in  the  result  of  the  following 
problem:  A  father  is  53  yrs.  old  and  his  son  28.  After 
how  many  years  will  the  father  be  twice  as  old  as  the  son  ? 

We  have  the  equation 

53  +  £c  =  2(28  +  ic), 
whence  cc  =  —  3. 

We  are  now  met  by  the  necessity  of 

(1)  interpreting  the  meaning  of  the  answer  —  3  yearn 
after  this  time,  or 

(2)  changing  the  statement  of  the  problem  so  as  to  avoid 
an  answer  which  seems  meaningless. 

It  is  immaterial  which  course  we  take.     We  may  say : 

(1)  —  3  years  after  this  time  shall  be  understood  to  mean 
3  years  before  this  time,  which  is  entirely  in  harmony  with 
our  interpretation  of  negative  numbers  (§  29) ;   or 

(2)  we  may  change  the  problem  to  read :  "  How  many 
years  ago  was  the  father  twice  as  old  as  the  son  ?  "  For 
this  latter  question  the  solution  would  be 

53 -a:  =  2(28 -a;). 
.•.x  =  3, 
and  the  answer  would  be  3  years  ago. 


SIMPLE   EQUATIONS.  177 

The  discussion  of  results  of  this  nature  is  well  illustrated 
in  an  ancient  problem  known  as  the  Problem  of  the  Couriers. 

A  courier,  A,  travels  at  the  uniform  rate  of  a  mi.  per 
hour  from  F ;  after  t  hrs.  a  second  one,  B,  starts  in  pursuit 
from  F  and  travels  at  the  uniform  rate  of  h  mi.  per  hour. 
After  how  many  hours  will  B  overtake  A  ? 


Pr- 


hx 
Solution.     1.    Let  x  =  the  number  of  hours  required. 
2.    Then  •.•  a  (^  +  x)  =  6x,  the  distance  B  must  travel, 

at  ,/-.    V  '^ 

(^■'  b-a 

Discussion.  1.  If  none  of  the  quantities  is  zero,  and  6  >  a,  the 
denominator  is  positive  and  .-.  x  is  positive. 

2.  But  lib  =  a,  the  denominator  is  zero  and  .-.  x  is  infinite  (§  170). 
I.e.,  if  they  are  traveling  at  the  same  rate  B  will  never  overtake  A. 

3.  And  if  b  <ja,  tlie  denominator  is  negative  and  .-.  x  is  negative. 
I.  e. ,  if  B  is  traveling  slower  than  A  he  will  never  overtake  him.  But 
if  the  problem  reads,  "  •  •  •  after  t  hrs.  B  passes  through  P  in  pursuit," 

then  the  result  would  mean  that  they  had  been  together    hrs. 

before  reaching  P.  \     —  a\ 

4.  If  either  t  =  0,  or  a  =  0,  the  numerator  is  zero  and  .-.  x  =  0, 
except  when  6  =  a,  in  whicli  case  x  is  undetermined  (§  168).  This  is 
evidently  true,  for  ift  =  0  and  they  are  traveling  at  tlie  same  rate  they 
will  always  be  together.  Or  if  a  =  0  and  a  =  6,  then  neither  courier 
is  traveling  at  all,  and  hence  they  are  always  together  at  P. 


EXERCISES.    LXXXVIII. 

Solve  the  following  and  discuss  the  results  according  to 
the  suggestions  given  above  and  in  the  problems. 

1.  A  bicyclist  starts  out  riding  10  mi.  per  hour,  and  is 
followed  after  30  mins.  by  a  second  riding  8  mi.  per  hour. 
In  what  time  will  the  second  overtake  the  first  ?     - 


178  ELEMENTS   OF  ALGEBRA. 

2.  A  bicyclist  starts  from  P,  riding  a  mi.  per  hour ;  after 
t  hrs.  another  follows  and  overtakes  him  in  h  hrs.  At  what 
rate  did  the  second  one  ride  ?     Discuss  for  ^  =  A  =  0. 

3.  A  bicyclist  starts  from  P,  riding  a  mi.  per  hour ;  he 
is  followed  after  t  hrs.  by  a  second  rider  traveling  c  times 
as  fast.  After  how  many  hours  will  the  second  overtake 
the  first  ?  Discuss  for  (1)  c  >  1,  t-^  0,  (2)  c  =  1,  t-^  0, 
(3)  c  <  1,  t^  0,    (4)  c  =  1,  t  =  0. 

4.  Two  trains  going  from  San  Francisco  to  Chicago,  on 
the  same  road,  pass  through  Omaha,  the  first  at  9.30  a.m., 
and  the  second  at  10  a.m.  The  first  train  travels  at  the 
rate  of  50  mi.  per  hour,  and  the  second  10%  slower.  At 
what  distance  from  Omaha  are  they  together  ? 

REVIEW    EXERCISES.    LXXXIX. 

1.  Are  x  =  2  and  x*  =  16  equivalent  equations  ?     Why  ? 

2.  Show  that  if  cc  is  a  factor  of  every  term  of  an  equa- 
tion, 0  is  a  root.     E.g.^  x^  -{-  2x  =  5x. 

3.  Solve  the  equation 


3a-2\a  +  3[a-2(a-a-2x)^\  =  lla. 

4.  Show  that  if  both  members  of  an  equation  have  a 
common  linear  factor  containing  the  unknown  quantity,  a 
root  can  be  found  by  equating  this  factor  to  zero. 

5.  What  is  the  fallacy  in  this  argument  ? 

1.  Let  X  =  a. 

2.  Then  x^  =  ax,  multiplying  by  x. 

3.  Then  x^  —  a^  =  ax  —  a^,  subtracting  a^. 

4.  Then  (x  -\-  a)  (x  —  a)  =  a  (x  —  a),  factoring. 

5.  .'.  2  a  (x  —  a)  =  a  (x  —  a),  hecsbuse  x-\- a  =  2a. 

6.  .*.  2  =  1,  dividing  by  a(x  —  a). 


CHAPTER   X. 

SIMPLE    EQUATIONS   INVOLVINCx   TWO  OR  MORE 
UNKNOWN   QUANTITIES. 

197.  A  single  linear  equation  containing  two  unknown 
quantities  does  not  furnish  determinate  values  of  these 
quantities. 

This  means  a  single  equation  in  which  the  similar  terms  have  been 
united.  I.e.,  x  +  y  =  x  -{-  3  is  not  included,  because  the  x's  have  not 
been  united. 

E.g.,  X  —  y  =  1  is  satisfied  if  x  =  1  and  y  =  0,  or  if  x  =  2  and  y  =  1, 
or  if  X  =  3  and  y  =  2,  etc. 

198.  But  two  linear  equations  containing  two  unknown 
quantities  furnish,  in  general,  determinate  values.  Simi- 
larly, as  will  be  seen,  a  system  of  three  linear  equations 

(ontaining  three  unknown  quantities,  a  system  of  four  linear 
^nations  containing  four  unknown  quantities,  •  •  •  a  system 
K  n  linear  equations  containing  n  unknown  quantities, 
urnish,  in  general,  determinate  values  of  all  of  these 
uantities. 


1 


199.    Equations  all  of  which  can  be  satisfied  by  the  same 
lues  of  the  unknown  quantities  are  said  to  be  simultaneous. 

E.g.,x-{-y  =  7,x  —  y  =  S,  are  two  equations  which  are  satisfied  if 
X  =  5  and  y  =  2.     Hence  they  are  simultaneous. 

But  x  +  y  =  7  and  x  -{-  y  =  S  cannot  be  satisfied  by  the  same  values 
of  X  and  y,  and  hence  they  are  not  simultaneous. 

The  equations  x  +  2  ?/  =  6,  Sx  -\-6y  =  9,  are  simultaneous  ;  but 
each  being  derivable  from  the  other  they  do  not  furnish  determinate 
values. 

179 


180  ELEMENTS   OF  ALGEBRA. 

I.     ELIMINATION   BY  ADDITION  OR   SUBTRACTION. 

200.  The  solution  of  two  simultaneous  equations  involv- 
ing two  unknown  quantities  is  made  to  depend  upon  the 
solution  of  a  single  equation  involving  but  one  of  the 
unknown  quantities.  The  usual  process,  by  addition  or 
subtraction,  is  seen  in  the  following  solutions : 

1.  Solve  the  system  of  equations 

L    4ic  +  3y  =  41. 

2.    Zx-2y  =  l. 

We  first  seek  to  give  the  ?/'s  coefficients  having  the  same  absolute 
values.  This  can  be  done  by  multiplying  both  members  of  the  first  by 
2,  and  of  the  second  by  3.     Then 

3.  8x  +  6y  =  82. 

4.  9x-62/  =  3. 

Add  equations  3  and  4,  member  by  member,  and 
6.  17x  =  85. 

6.  .-.  X  =  5. 
Substitute  this  value  in  equation  1,  and 

7.  4.5  +  3y  =  41. 

8.  .-.  32/ =  21. 

9.  .-.  2/ =  7. 

Check.  Substitute  these  values  in  equation  2  (because  2/was  obtained 
by  substituting  in  equation  1),  and  3  •  5  —  2  •  7  =  1. 

For  brevity  we  shall  hereafter  use  the  expressions,  in 
solutions,  "Multiply  2  hj  5,"  etc.,  meaning  thereby,  "Multi- 
ply both  members  of  equation  2  by  5,"  etc. 

201.  When  one  of  the  unknown  quantities  has  been  made 
to  disappear  (as  in  passing  from  steps  3  and  4  to  step  5 
above)  it  is  said  to  be  eliminated. 

In  the  above  solution  y  was  eliminated  by  addition.  The 
quantity  x  may,  however,  be  eliminated  first,  by  subtraction, 
as  in  the  following  solution. 


SIMPLE   EQUATIONS. 


181 


2.  Solve  the 

system  of  equations 

1.  4ic  +  32/  =  41. 

2.  3x-2y  =  l. 

3.  ••• 

12x  +  Oy  =  123,  multiplying  1  by  3, 

4.  and 

12x-82/  =  4. 

(Why  ?) 

|5.  . 

11  y  =  119,  subtracting  4  from  3. 

^.... 

2/ =  7. 

(Why  ?) 

^7.  .-. 

4a; +  21  =41. 

(Why  ?) 

8.  .-. 

4x  =  20. 

(Why?) 

«.  .-. 

x  =  b. 

(Why  ?) 

Check.     In  which  equation  should  these  values  now  be  substi- 
tuted ?     (Why  ?) 


Other  types  are    illustrated   in  the  two  problems  fol- 
lowing, 

3.  Solve  the  system  of  equations 

^  V  r. 

1. =  2. 

3      2 

X      y      „ 
2^4 

It  is  not  worth  while  here  to  clear  of  fractions.     Simply  multiply 
)th  members  of  the  first  by  i,  and 

--1  =  1 
6      4 

2x 
».   .*.  -—  =  8,  adding  2  and  3. 

6.    .-.  X  =  12.  (Why  ?) 

It  is  now  apparent  that  y  can  easily  be  found  and  the  results  checked 
in  the  usual  way. 

I 


182  ELEMENTS   OF   ALGEBRA. 

4.    Solve  the  system  of  equations 

X       y      4t 

X       y 

These  are  not  linear  equations  because,  when  cleared  of  algebraic 
fractions,  they  are  of  the  second  degree.  But  they  can  easily  be 
solved  by  the  methods  of  linear  equations  as  here  suggested. 

3.  \^l  =  ^-  ^^^y^) 

4.  .'.  -  =  -,  subtracting  1  from  3. 

X      4 

5.  .-.  4  =  »,  multiplying  by  4  x. 
Hence,  y  is  easily  found  to  be  2,  and  the  results  check. 


EXERCISES.    XC. 

Solve  the  following,  checking  each  result  by  proper  sub- 
stitution : 


9. 


Ix 

-32/  = 

:3. 

5x 

+  7)/  = 

:25. 

x-\- 

■17  2/  = 

53. 

Sx  +  y  = 

19. 

6x-5y: 

-12. 

12  a; -11 2/ 

=  27. 

X 

7      2 

X 

2  + 

F^^ 

3 

X 
X 

4_ 

y~ 

5      31 

y~  3 

15 

2  ' 

5. 


2. 

3x  +  5y  =  5. 

4,x-3y  =  26., 

4. 

5x  +  2y  =  l. 

18x-{-8y  =  ll. 

6. 

l.Tx-{-l.ly  =  13, 

1.3cc-0.l2/  =  l. 

8. 

5  +  10-^- 

^  4.2/-S 
10  +  5"^' 

0. 

¥-1='' 

i+ir-»- 

.    ELIMINATION  BY  SUBSTITUTION  AND  COMPARISON. 

202.  After  finding  the  value  of  one  unknown  quantity 
addition  or  subtraction  the  other  is  usually,  but  not 

necessarily,  found  by  substitution.  It  is  often  more  con- 
venient to  find  each  by  substitution,  especially  when  one 
of  the  coefficients  is  1. 

This  method  of  elimination  by  substitution  is  illustrated 
in  the  following  solution  : 

1.  Given  x  — |?/=— 5, 

2.  and  3  x  +  2  ?/  =  45. 
From  equation  1  we  have  : 

3.  X  =  f  ?/  -  5.  (Why  ?) 
Substitute  this  value  in  equation  2,  and 

4.  2  y  —  15  +  2  2/  =  45,  from  which 
4  ?/  =  60. 

6.   .-.  y  =  15. 

From  this  x  is  found,  by  substitution,  to  be  5,  and  the  results  check. 

It  is  not  necessary  that  the  coefficient  oi  x  oi  y  should 
1,  although  this  is  the  case  in  which  the  method  is  most 
requently  employed.     Consider,  for  example,  the  follow- 
ing solution  : 

1.  Given  2x-\-by=  154, 

2.  and  30x-2?/  =  0. 

3.  From  equation  2,  x  =  ^T^y. 

4.  Substituting,  -^^y  -\-  ^y  —  154, 
lence  y  =  20. 

x  =  2. 

203.  A  special  form  of  substitution  occurs  when  the  value 
one  of  the  unknown  quantities  is  found  in  each  equation, 

nd  these  values  are  compared.     This  is  called  elimination 
by  comparison. 


i 


184  ELEMENTS   OF  ALGEBRA. 

The  method  is  illustrated  in  the  following  solution : 

1.  Given  x  —  ^y  =  —  5, 

2.  and  3  x  +  2  y  =  45. 
Solving  equations  1  and  2  for  x,  we  have  : 

3.  x  =  f2/-5, 

4.  and  x=  15  —  f  y. 

Substituting  the  value  of  x  from  step  3  in  step  4,  or,  what  is  the 
same  thing,  comparing  the  values  of  x  (by  ax.  1),  we  have  : 

5.  f?/-5  =  15-^y. 

6.  .-.  f2/  =  20.                                          (Why?) 

7.  .-.  2/ =  15.                                        (Why?) 

8.  .-.  X  =  5,  by  substituting  in  step  3. 
Check.  Substituting  in  both  of  the  original  equations, 

5  _  1 .  15  =  _  6. 

3  •  5  +  2  ■  15  =  45. 

EXERCISES.    XCI. 

Solve  the  following  by  substitution  or  comparison,  check- 
ing the  results  as  usual : 


1. 

x+y=:ll. 

2. 

X  +  y  =  s. 

3  £c  +  2  2/  =  44. 

X  —  y  =:  d. 

3. 

x  =  y. 

4. 

X  -\-ay  —  h. 

3ic  +  5?/  =  120. 

ex  -\-  y  =■  d. 

5. 

x-y-l  =  0. 

6. 

ax  +  hy  =  c. 

2x  +  y-29  =  0. 

a'x  +  b'y  =  c'. 

7. 

x  +  y  =  6912. 

8. 

x-\-2y  =  30. 

X  =  4444  -f  y. 

i^-i2/  =  3. 

9. 

x-\-lly  =  300. 

10. 

x-\-l^y  =  26^^^. 

llx-y  =  104.. 

4|2/-^  =  44|. 

11.    1.543689  cc 

-y 

=  1.543689. 

aj- 0.839286  7/ 

=  0.839286. 

SIMPLE   EQUATIONS.  185 


III.     GENERAL  DIRECTIONS. 

204.    The  following  general  directions  will  be  found  of 
ome  value,  although  the  student  must  use  his  judgment 
in  each  individual  case. 

1.  If  the  equations  contain  symbols  of  aggregation,  decide 
whether  it  is  better  to  remove  them  at  once. 

It  is  usually  best  to  remove  them,  as  in  a  case  like  ex.  18,  p.  188. 
But  in  a  case  like  ex.  17,  p.  188,  it  is  evidently  better  to  add  at  once. 

2.  If  the  equations  are  in  fractional  form,  decide  whether 
it  is  better  to  eliminate  without  clearing  of  fractions. 

See  pp.  181,  182,  illustrative  problems  3,  4.  Much  time  is  often 
wasted  by  clearing  of  fractions  unnecessarily.  This  is  also  seen  in 
the  example  on  p.  193. 

3.  If  it  seems  advisable,  clear  of  fractions  and  reduce 
each  to  the  form  ax  +  by  =  c. 

See  illustrative  problem  1,  p..  186.  The  same  course  will  naturally 
be  followed  with  an  example  like  ex.  8,  p.  187. 

4.  If  the  coefficient  of  either  unknown  quantity  is  1,  it 
is  usually  advisable  to  eliminate  by  substitution. 

See  illustrative  problem  1,  p.  186,  steps  4,  6,  7.  This  is,  however, 
not  often  the  case. 

5.  Otherwise  it  is  generally  best  to  eliminate  by  addition 
or  subtraction. 


This  is  the  plan  usually  employed. 


I' 
6.  If  the  unknown  quantity  is  in  an  exponent,  follow  the 
Ian  suggested  in  §  205. 
It  is  here  assumed  that  the  root  of  the  single  equation 
erived  from  the  two  given  equations  satisfies  those  equa- 


186  ELEMENTS   OF  ALGEBRA. 

Illustrative  problems.     1.  Solve  the  system  of  equations 

y 

2.  ?  +  5  =  ^  +  2. 

X  X 

Here  it  is  not  best  to  attempt  to  eliminate  without  clear- 
ing of  fractions.     Multiplying  both  numbers  of  1  by  y, 


3. 

l+x  +  3?/  =  5?/. 

Ax.  6 

4.    .-. 

x  =  2y  -\. 

(Why  ?) 

5. 

2  +  5x  =  ?/  +  2x,  from  2. 

(Why?) 

6.    .-. 

3  X  —  ?/=:—  2,  from  5. 

(Why  ?) 

7.    .-. 

6^/  —  3  —  ?/=—  2,  -substituting  4  in  6. 

8.    .-. 

'oy=\. 

(Why  ?) 

9.    .-. 

1        .>          -3 

(Why?) 

Check. 
d 

Substituting  in  both  given  equations, 
2  +  3  =  5,  from  1, 
-i/  +  5  =  -i+-2, 
f  =  f ,  from  2. 

and 


205.  Equations  in  which  the  unknown  quantities  appear 
as  exponents  are  called  exponential  equations. 

Exponential  equations  of  the  following  type  are  easily 
solved  by  means  of  simple  equations. 

2.  Solve  the  system  of  equations 


60 


1.  •.• 

a2a;.a3|/  =  a32, 

2.    .-. 

Q-ix  +  Zy  —  a^. 

3.    .-. 

2x  +  3?/  =  32. 

4.    Similarly, 

3x  +  42/  =  44. 

Solving, 

X  =  4,  2/  =  8. 

SIMPLE   EQUATIONS. 


187 


EXERCISES.    XCII. 

Solve  the  following,  checking  as  usual : 


=  V^. 


3. 


X 


m      n 

=p. 

X       y 

5.   p^""  -jy"-'  =f'^. 
9.    ^"•^:65  2/^^20_ 

11.      a^  +  TV^/^^l. 

-   1  cc  =  61. 


2/ 


ri3.  ^+^  =  «. 

a;  — 2/H-l 


y  —  X  -{-1 
x  —  y  +  1 


ab. 


L5.    ^'  +  ^ 


6.3. 


3+56   ~'^*^''^- 


2. 


-4-^  =  1 


a       b 


X 

m 

3a; 
~5~ 


6. 


3      8~^' 

4^ .  16^  =  2^°. 
16^.22  2/  =41 


4a;  +  81 
10^/- 17 
12a; +  97 


152/ -17 

10.    7  a; -1 7/3=48. 
5  ?/  +  I  a;  =  26. 

12.    17  a; -13?/ =.144. 
23  a;  +  19  2/  =  890. 


14. 


16. 


+  1  =  0. 
X      y 

: — I =  a—  0. 

X  V 


x-y 


15 


9x 


32/ +  44 


too. 


188  ELEMENTS  OF  ALGEBllA. 

17.    a{x  -\-  y)  —  h{x  —  y)  =  2  a. 
a{x  —  y)  —  b {x  -\-  y)  =  2h, 


18.    10[ic  +  9(3/-8ic  +  7)]=6. 
5[a^  +  4(2/-3^T2)]=l. 


19. 

a 

-x-y  = 

a  —  c          a 

a  + 

6          a  +  1 

c^- 

c       a 

20. 

5y 
6 

42/-ig 

3 

1      03      20  - 
6"^        3 

22/. 

X 

6       +^ 

3 

21. 

2a;_ 
1/   " 

29 
14' 

2/  +  4CC  +  6  = 

4^/^  +  13^7/ 

-12ic2 

42/-3X 

-1 

22. 


7       +            3 

^     = 

=  a;  + 

2/- 

5i. 

11— £c      Ax  -^8y  - 
.2        '              9     ^ 

—  2 

=  8- 

(y- 

-:r). 

4:x^-^2xy-h288- 

6  2/^ 

=  2x 

+  3 

2/- 

131, 

2^  +  13-22/ 

5a;- 

-42/: 

^22. 

23. 


24.    7y  +  13-5.  _3y  +  2x-16. 

4  ^  3 

,5y  +  2x      3.T-12  +  8y       ,       15 +  2?/ -4a; 


SIMPLE   EQUATIONS.  189 

IV.    APPLICATIONS  OF  SIMULTANEOUS   LINEAR  EQUA- 
TIONS  INVOLVING  TWO  UNKNOWN  QUANTITIES. 

Illustrative  problem.  The  sum  of  two  numbers  i$  12 
and  7  times  the  quotient  of  one  divided  by  the  other  is  5. 
Required  the  numbers. 


1.  Let 

2.  Then 

X,  ?/  =  the  numbers. 
x  +  y  =  12,  and 

3. 

7  •  -  =  5,  by  the  conditions  of  the  problem. 

4.  .-. 

5.  And 

6.  .-. 

7.  .-. 

8.  .-. 

y  =  12  —  X,  from  2. 
7  X  =  5  2/,  from  3* 
7x  =  60-5x. 
12x  =  60,  andx  =  5. 

y  =  7,  from  step  4. 

Ax.  3 

Ax.  6 

(Why  ?) 

(Wliy?) 

Check.     The  sum  of  5  and  7  is  12,  and  7  times  f-  is  5. 
EXERCISES.    XCIII. 

1.  The  sum  of  two  numbers  is  30  and  their  difference 
is  17.     Eequired  the  numbers. 

2.  What  is  that  fraction  which  equals  -J  when  1  is 
added  to  the  numerator,  but  equals  ^  when  1  is  added  to 
the  denominator  ? 

3.  A  number  of  two  figures  is  5  times  the  sum  of  its 
ligits.     If  9  is  added  to  the  number,  the  order  of  its  digits 

reversed.     Required  the  number. 

4.  A  man  invests  $16,000  for  8  yrs.  and  $11,000  for 

yrs.,  and  receives  from  the  two  $8090  interest.    Had  the 

[•st  been  invested  at  the  same  rate  as  the  second  and  the 

icond  at  the  same  rate  as  the  first,  he  would  have  received 

f310  more  interest  in  the  same  times.     Required  the  rate 

tt  which  each  was  invested. 


190  ELEMENTS  OF  ALGEBKA. 

5.  The  sum  of  two  numbers  is  s  and  their  difference 
is  d.  Required  the  numbers.  From  the  result,  deduce  a 
rule  for  finding  two  numbers,  given  their  sum  and  their 
difference. 

6.  The  sum  of  two  capitals,  each  invested  at  5%,  is 
$12,000,  and  the  sum  of  5  yrs.  simple  interest  on  the 
larger  and  4  yrs.  simple  interest  on  the  smaller  is  $2800. 
Required  the  capitals. 

7.  Divide  the  two  numbers  80  and  90  each  into  two 
parts  such  that  the  sum  of  one  part  of  the  first  and  one 
part  of  the  second  shall  equal  100,  and  the  difference  of 
the  other  two  parts  shall  equal  30. 

8.  Two  points  move  around  a  circle  whose  circumfer- 
ence is  100  ft. ;  when  they  move  in  the  same  direction  they 
are  together  every  20  sees. ;  when  in  opposite  directions 
they  meet  every  4  sees.     Required  their  rates. 

9.  The  boat  A  leaves  the  city  C  at  6  a.m.  ;  an  hour 
later  the  boat  B  leaves  the  city  D,  80  mi.  from  C,  and 
meets  A  at  11  a.m.  They  would  also  meet  at  11  a.m.  if  B 
left  at  6  A.M.  and  A  45  mins.  later.     Required  their  rates. 

10.  Of  two  bars  of  metal,  the  first  contains  21.875% 
pure  silver  and  the  second  14.0675%.  How  much  of  each 
kind  must  be  taken  in  order  that  when  melted  together 
the  new  bar  shall  weigh  60  oz.,  and  18.75%  shall  be  pure 
silver  ? 

11.  A  marksman  fires  at  a  target  500  yds.  distant  and 
hears  the  bullet  strike  4-J  sees,  after  he  fires ;  an  observer 
standing  400  yds.  from  the  target  and  650  yds.  from  the 
marksman  hears  the  bullet  strike  2-^  sees,  after  he  hears 
the  report.  Required  the  velocity  of  sound  and  the 
velocity  of  the  bullet,  each  supposed  to  be  uniform. 


SIMPLE   EQUATIONS.  191 

12.  Find  two  numbers  the  sum  of  whose  reciprocals 
is  5,  and  such  that  the  sum  of  half  of  the  first  and  one- 
third  of  the  second  equals  twice  the  product  of  the  two 
numbers. 

13.  Two  bodies  are  96  yds.  apart.  If  they  move 
)wards  each  other  with  uniform  (but  unequal)  rates,  they 
rill  meet  in  8  sees. ;  but  if  they  move  in  the  same  direc- 

y^on  the  swifter  overtakes  the  slower  in  48  sees.    Required 
le  rate  of  each. 

14.  The  sum  of  two  numbers,  one  of  one  figure  and  the 
ither  of  five  figures,  is  15,390.     Writing  the  fi^rst  number 

the  first  digit  to  the  left  of  the  second  number  gives  a 
lumber  4  times  as  large  as  that  which  is  obtained  by  writ- 
ig  it  as  the  last  digit  to  the  right.     Required  the  numbers. 

15.  A  reservoir  has  two  contributing  canals.  If  the 
first  is  open  10  mins.  and  the  second  13  mins.,  15  cu.  yds. 
of  water  flow  in ;  if  the  first  is  open  14  mins.  and  the 
second  5  mins.,  2.4  cu.  yds.  more  flow  in.  How  many 
cubic  yards  of  water  per  minute  are  admitted  by  each  ? 

16.  A  silversmith  has  two  silver  ingots  of  different 
quality.  He  melts  13  oz.  of  the  finer  kind  with  12  oz.  of 
the  other,  the  resulting  ingot  being  852  fine  (see  p.  169, 
ex,  44)  ;  but  if  he  melts  1.5  oz.  of  the  finer  kind  with 
1  oz.  of  the  other  the  resulting  ingot  is  860  fine.    Required 

le  fineness  of  each  original  ingot. 

17.  It  is  shown  in  physics  that  if  a  body  starts  with  a 
relocity  of  u  ft.  per  second,  and  if  this  velocity  increases 

ft.  per  second,  then  at  the  end  of  t  sees,  the  body  will 
Lve  passed  over  ut  +  ^ft^.      Suppose  /  is  uniform  and 
lat  in  the  11th  and  15th  sees,  the  body  passes  through 
J4  ft.  and  32  ft.,  respectively,  find  u  and  /. 


192  ELEMENTS  OF  ALGEBRA. 

V.  SYSTEMS  OF  EQUATIONS  WITH  THREE  OR  MORE 
UNKNOWN  QUANTITIES. 

206.  In  general,  three  linear  equations  involving  three 
unknown  quantities  admit  of  determinate  values  of  these 
quantities.  For  one  of  the  quantities  can  be  eliminated 
from  the  first  and  second  equations,  and  the  same  one  from 
the  first  and  third,  thus  leaving  two  linear  equations  involv- 
ing only  two  unknown  quantities.  Similarly  for  a  system 
of  four  linear  equations  containing  four  unknown  quanti- 
ties, and  so  on. 

Illustrative  problems.  1.  Solve  the  following  system  of 
equations : 

1.  5x  —  Sy-{-4:Z==17. 

2.  2x  +  7y  —  5z  =  5. 

3.  9x-2y-z  =  S. 

We  first  proceed  to  elimiuate  z  from  1  and  2. 

4.  25x  -  15  2/  +  20  z  =  85,  multiplying  1  by  5. 

5.  8  a  -F  28  ?/  -  20  z  =  20,  multiplying  2  by  4. 

6.  .-.  33x  +  132/=  105.  (Why?) 
We  now  proceed  to  eliminate  z  from  1  and  3. 

7.  36  X  -  8  y  -  4  2  =  32,  multiplying  3  by  4. 

8.  .-.  '  41 X  -  11  ?/  =  49,  from  1  and  7. 
We  now  proceed  to  eliminate  y  from  6  and  8. 

9.  363  X  +  143  2/  =  1155,  multiplying  6  by  11. 

10.  533  X  -  143  y  =  637,  multiplying  8  by  13. 

11.  .-.  896x  =  1792.                                   (Why?) 

12.  .-.  x  =  2.                                         (Why?) 

13.  .♦.  2/  =  3,  substituting  in  6. 

14.  .-.  2  =  4,  substituting  in  1. 
Cfieck.  Substitute  in  2  and  3.     (Why  not  in  1  ?) 

4  +  21  -  20  =  5,  and  18  -  6  -  4  =  8. 


f 


SIMPLE   EQUATIONS. 


2.  Solve  the  following  system  of  equations 
ox       1  y       9z 


2.     ?+? 
X  y 


67. 


5  +  £ 

y  z 


38. 


193 


We  first  proceed  to  eliminate  -  from  1  and  2. 


4         4         4 

\ 1 =  4,  from  1. 

5x       ly      92 


2,3         4        67     , 

—  — ,  from  2. 

9x      92/      9z       9 


4 

li  +  ^  =  103. 
5x      72/ 


(Why  ?) 
(Why  ?) 
(Why  ?) 


We  then  proceed  to  eliminate  -  from  2  and  3. 


7. 

u 
X 

i^  +  "  =  _  76,  from  3. 
2/       z 

(Why  ?) 

8.    .-. 

^  -  ^-  =  -  9,  from  2  and  7. 

(Why  ?) 

We  then  proceed  to  eliminate  -  from  6  and  8. 

8. 

322 
5x 

+  ^  =  721,  from  6. 
72/ 

(Why  ?) 

L 

456 

7x 

399            513     ^ 

—    —  = ,  from  8. 

72/              7 

(Why  ?) 

11. 

4534      4534    ^         ,,       ,  ,_ 
=          ,  from  9  and  10. 
35x          7 

(Why  ?) 

12.    .'. 

^       ^        A          1 

=  5,  and  x  =  -• 

X                           5 

(Why  ?) 

13.    .-. 

p7,  and2/  =  -. 

(Why  ?) 

14.    .-. 

l=-9,and.=.-^. 

(Why  ?) 

194  ELEMENTS   OE  ALGEBRA. 

Check.     Substitute  in  1  and  2.     (Why  not  in  3  ?) 
1  +  1-1  =  1. 
10  +  21+36  =  67. 

The  equations  in  ex.  2  are  not  linear  in  x,  y,  z  (why  ?), 
and  it  is  unwise  to  clear  of  fractions  (why  ?).     The  equa- 
tions are  linear  in  -?  ->  -?  and  it  is  better  to  solve  as  if 
X    y    z 

these  were  the  unknown  quantities. 

3.  Solve  the  following  system  of  equations  : 

1.  ic  4-  2/  —  s;  =  6. 

2.  ic4-2/  +  2«  =  —  3. 

3.  x-2y  -z  =  ^. 

Frequently  systems  of  equations  offer  some  special  solu- 
tion, as  in  this  case. 

Adding  the  equations,  member  by  member, 

4.  3x  =  3. 

5.  .-.  x  =  l. 
Subtracting  2  from  1,  member  by  member, 

6.  -32  =  9. 

7.  .-.  2  =  -  3. 

8.  .-.  2/  =  2,  substituting  in  1. 
Check.     Substitute  in  2  and  3. 

1  +  2-6=  -3. 
1-4  +  3  =  0. 

4.  Solve  the  following  system  of  equations  : 

1.  w-{-2x-\-y  — z  =  4^. 

2.  2w-\-x-\-y-\-z  =  l. 

3.  ^w  —  x-\-2y  —  z  =  l. 

4.  4.W  +  3x-y-\-2z  =  l^. 


SIMPLE   EQUATIONS. 


196 


Eliminating  z  from  1  and  2, 

5.  3w)  +  3a;  +  22/  =  ll. 
Also  from  1  and  3, 

6.  2«;-3x  +  y=-3. 
Also  from  1  and  4, 

7.  6iw4- 7a;  +  y  =  21. 
Eliminating  y  from  6  and  6, 

8.  to-9x=-17. 

9.  Also  from  6  and  7,  2  w>  +  5  cc  =  12. 
Eliminating  w  from  8  and  9, 

10.  jc  =  2. 

11.  .-.  io  =  1,  substituting  in  8. 

12.  .-.  2/  =  1,  substituting  in  6. 

13.  .-.  2  =  2,  substituting  in  1. 
Check.     Substitute  in  2,  3,  and  4.     (Why  not  in  1  ?) 

2+2  +  1+2  =  7. 
3-2  +  2-2  =  1. 
4  +  6-1+4=  13. 


EXERCISES.    XCIV. 

Solve  the  following  systems  of  equations 


1.  l+i=l. 

X       y 

2. 

|  +  f  +  |  =  258. 

^  +  ^2. 

X       z 

M+5-^«^- 

y^  z      2 

M  +  f  =  296. 

3.       lx-^y  = 

1. 

4. 

5x  —  6y-\-4:Z  =  15 

llz-7u  = 

1. 

7x  +  4:y-3z  =  19 

4rz-7y  = 

1. 

x  +  y  =  l. 

19x-3u^ 

1. 

a;  +  6  «  =  39 

196  ELEMENTS   OF  ALGEBRA. 

5.    x  +  y  =  16.  6.    x-\-y-z  =  132. 

z-{-x  =  22.  x-y  +  z  =  65.4. 

y  +  z=2S.  -x  +  y  +  z  =  - 1.2. 

7.    a*  •  a^  +  2  ==  a>\  8.    a^x  +  h^y  +  Ci-t;  =  ^i. 

~    a^'  ■  a^  +  ^  =  a^"^.  a^x  +  h^y  +  Cg;^  =  c^g. 

9.    a;  =  21— 4?/.  10.              ic  +  y  +  ;^  =  5. 

z  =  9-^x,  3x-5y  +  7z  =  75. 

y  =  64  -  7^5:.  9ic  -  11  ^  +  10  =  0. 


11. 


X         y 

L8  =  i8. 

a:         ?/        ^ 

X        y 

^  =  25. 

z 

13. 

3y-l 
4 

5.-^4;*; 
4+3 

3a;  +  l        ^        1 

12.    7ic  —  2;t; +  3ii  =  17. 
4?/  —  2«  +  V  =  11. 
5?/-3x-2i^  =  8. 
4  7/  -  3  w  +  2  V  =  9. 
3z  +  8it  =  33. 


6  ^      .T      9 

T~2"^5' 


14  "^  6       21  "^  3 


14.  «4^+3?/  +  2.    ^3x  +  y  +  2     __  ^2«  +  lo_ 

15.  2 1^;  +  cc  -  10  ?/  +  0.5  ^  =  7.62. 

3  ^  -  2  .X  +  2  ?/  +  3  s;  =  8.26. 
w  +  3x  +  5y  —  z  =  8.61. 
-6?^-2ic  +  32/+  10^  =  25.51. 


SIMPLE   EQUATIONS. 


197 


VI.     APPLICATIONS   OF   SIMULTANEOUS   LINEAR   EQUA- 
TIONS  INVOLVING   THREE   UNKNOWN   QUANTITIES. 

Illustrative  problem.  A  certain  number  of  three  figures 
such  that  when  198  is  added  the  order  of  the  digits  is 

jversed  ;  the  sum  of  the  hundreds'  digit  and  the  tens'  digit 
the  units'  digit ;    and  the  number  represented  by  the  two 

jft-hand  digits  is  4  times  the  units'  digit.  Required  the 
dumber. 

1.  Let  a  =  the  hundreds'  digit,  6  the  tens',  c  the  units'. 

2.  Then  100  a  +  10  6  +  c  =  the  number. 

3.  Then,  by  the  first  condition, 
100a  +  10&  +  c  +  198  =  100c  +  10  6  +  a. 

4.  By  the  second  condition,    a  -\-h  =  c. 

5.  By  the  third  condition,  10  a  +  6  =  4  c. 

6.  .-.  the  equations  are 
a  —  c  =  —  2,  from  step  3. 

a  +  &  —  c  =  0,  from  step  4. 
10a  +  &  —  4c  =  0,  from  step  5. 

7.  Solving,  a  =  1,  &  =  2,  c  =  3. 

.-.  the  number  is  123. 

Check  by  noting  that  123  answers  all  of  the  conditions  of  the  origi- 
nal  statement. 


EXERCISES.    XCV. 

1.  What  three  numbers  have  the  peculiarity  that  the 
pm  of  the  reciprocals  of  the  first  and  second  is  ^,  of  the 
irst  and  third  ^,  and  of  the  second  and  third  ^  ? 

2.  There  is  a  certain  number  of  six  figures,  the  figure 
units'  place  being  4;  if  this  figure  is  carried  over  the 

ther  five  to  occupy  the  left-hand  place,  the  resulting 
lumber  is  four  times  the  original  one.  Required  the  origi- 
lal  number. 


108  ELEMENTS  OE  ALGEBRA. 

3.  Divide  the  number  96  into  three  parts  such  that  the 
first  divided  by  the  second  gives  a  quotient  2  and  a 
remainder  3;  and  the  second  divided  by  the  third  gives 
a  quotient  4  and  a  remainder  5. 

4.  The  middle  digit  of  a  certain  number  of  three  figures 
is  half  the  sum  of  the  other  two ;  the  number  is  48  times 
the  sum  of  the  digits.  Subtracting  198  from  the  number, 
the  order  of  the  digits  is  reversed.      Required  the  number. 

5.  Of  3  bars  of  metal,  the  first  contains  750  oz.  silver, 
62^  oz.  copper,  187-|-  oz.  tin;  the  second,  62|-  oz.  silver, 
750  oz.  copper,  187^  oz.  tin  ;  and  the  third  no  silver,  875  oz. 
copper,  125  oz.  tin.  How  many  ounces  from  these  bars 
must  be  melted  together  to  form  a  bar  which  shall  contain 
250  oz.  silver,  562|-  oz.  copper,  and  187^  oz.  tin  ? 

6.  Of  three  bars  of  metal,  the  first  contains  750  oz. 
silver,  200  oz.  copper,  50  oz.  tin ;  the  second,  800  oz.  silver, 
125  oz.  copper,  75  oz.  tin ;  and  the  third  700  oz.  silver, 
250  oz.  copper,  50  oz.  tin.  How  many  ounces  from  these 
bars  must  be  melted  together  to  form  a  bar  which  shall 
contain  765  oz.  silver,  175  oz.  copper,  and  60  oz.  tin  ? 

7.  Two  bodies,  A  and  B,  start  at  the  same  time  from  the 
points  P  and  Q,  respectively,  and  move  at  uniform  rates 
towards  one  another,  B  faster  than  A;  at  the  end  of 
18  sees.,  and  again  at  the  end  of  30  sees.,  they  are  48  ft. 
apart.  Had  they  moved  in  the  same  direction,  B  follow- 
ing A,  at  the  end  of  40  sees,  they  would  have  been  48  ft. 
apart.     Determine  their  rates  and  the  distance  PQ. 

Solutions  by  determinants.  The  treatment  of  simulta- 
neous linear  equations  by  determinants  is  set  forth  in 
Appendix  VII,  and  should  be  taken  at  this  point  if  time 
allows. 


SIMPLE   EQUATIONS.  199 

REVIEW  EXERCISES.    XCVI. 

1.  Solve  the  equation  2.25  x  —  5  —  0.4a;  +  2.6  =  2a;  — 3. 

2.  By  the  Remainder  Theorem  ascertain  whether  10  x^ 
—  13  ic^  —  5  a;  +  3  is  exactly  divisible  by  2  a;  —  3. 

3.  Form  an  integral  linear  function  of  x  which  shall, 
equal  37  when  a;  =  10,  and  4  when  x  =  —  1. 

4.  Form  an  integral  linear  function  of  x  which  shall 
vanish  when  a;  =  2,  and  which  shall  equal  4  when  x  =  3. 
(If /(a;)  =  7nx  -f-  n,  then  2  m  -\-  n  =  0  and  3  m  -{-  n  =  A.) 

5.  Show  that  the  following  set  of  equations  are  not 
simultaneous  and  hence  cannot  be  solved : 

62x  +  93tj  =  31. 
2x-\-3y  =  4:. 

-i-i  +  1 

6.  Simplify  -'  +  -+^-'      - 

a^-lH--    aj  +  lH-- 

X  X 

7.  Write  down  by  inspection  the  quotient  of 

-3  +  -^2  +  —  +  4  by  4  +  -  +  1.     Check. 

*Aj  %K/  %Mj  *Aj  %aj 

13      2  2       3 

8.  Multiply  -1  +  -^ ^-7by  — f-4by  detached 

JLf  JL  JU  *kj  *Aj 

)efficients.     Check. 

^.   .^1       3?/  ,  5^/'      52/«  ,37/*        .  ,       1      y 

9.  Divide  — r  +  ^ ^  +  -^-2/    ^y  -^-- 

x^       x^         x^         x^  X        ^        _    x^      X 

y^  by  detached  coefficients.     Check. 
10.    Solve  the  equation 

2a;-l      2a;  +  5      2a;  +  l      ^^  +  3^^ 

2a;  +  l       2a;  +  7       2a;  +  3      2a;  +  5 


200  ELEMENTS   OF   ALGEBRA. 

11.    Solve  the  system 

0.2x-{-0.3y-hOAz  =  25. 
0.3x  +  0.7y-{-0.6z  =  4.5. 
0.4a; +  0.8v/ +  0.9^  =  58. 


12.    Solve  the  system 

1      1      1_1      3      2_3      1_ 

xyzxyzxy 

3 
4 

13.    Solve  the  system 

3x-5y  +  4.z  =  0.5. 

7x-{-2y-3z  =  0.2. 

4:x  -\-3y  —  z  =  0.7. 

14.    Solve  the  system 

x-^y-\-z  =  3.824. 

1.25  a;  +  23.8  y  +  3.1z  =  7.5276. 

1.1^  +2y-0.5z  =  l.S505. 

15.  The  sum  of  three  capitals  is  $111,000.  The  first  is 
invested  at  4%,  the  second  at  4:|-%,  and  the  third  at  5%, 
and  the  total  annual  interest  is  $5120.  If  the  first  had 
been  invested  at  2^(^q,  the  second  at  3%,  and  the  third  at 
4%,  the  total  annual  interest  would  have  been  $3710. 
Eequired  the  capitals. 

16.  In  each  of  three  reservoirs  is  a  certain  quantity  of 
water.  If  20  gals,  are  drawn  from  the  first  into  the  second, 
the  second  will  contain  twice  as  much  as  the  first ;  but  if 
30  gals,  are  drawn  from  the  first  into  the  third,  the  third 
will  contain  20  gals,  less  than  4  times  as  much  as  the  first ; 
but  if  25  gals,  are  drawn  from  the  second  into  the  third, 
the  third  will  contain  50  gals,  less  than  3  times  the  second. 
How  many  gallons  does  each  contain  ? 


CHAPTER    XI. 
INDETERMINATE    EQUATIONS. 

207.  A  linear  equation  involving  two  unknown  quantities 
can  be  satisfied  by  any  number  of  values  of  those  quantities. 

E.g.,  in  the  equation  x  -{■  y  =  b 

can  equal  •  •  •  -  2,  -  1,  0,  1,  2,  3,  4,  5,       6,  •  •  • 

16  corresponding  values  of  y  being    7,       6,  5,  4,  3,  2,  1,  0,  —  1,  •  •  •. 
But  of  course  this  applies  only  to  equations  after  like  terms  are 
:  united,  and  not  to  an  equation  like  x  -^  y  =  x  -\-  "1. 

208.  Equations  like  the  above,  which  can  be  satisfied  by 
an  unlimited  number  of  values  of  the  unknown  quantities 
are  called  indeterminate  equations. 

209.  Since  two  equations  containing  three  unknown 
quantities  give  rise,  by  eliminating  one  of  these  quantities, 
to  a  single  equation  containing  only  two,  it  follows  that, 
in  general,  Two  equations,  each  containing  three  unknown 
quantities,  are  indeterminate  as  to  all  of  these  quantities. 

E.g.,  the  two  equations 

2x  +  3  2/  +  2  =  10, 

3x  +  22/  +  2  =  8, 

give  rise  to  the  single  equation 

-  X  +  2/  =  2, 

or  to  by  -\-  z  =  14, 

or  to  5  X  +  2  =  4, 

all  three  of  which  are  indeterminate. 

201 


202  ELEMENTS   OF  ALGEBRA. 

210.  Similarly,  it  is  evident  that,  in  general,  n  linear 
equations,  each  containing  n  +  1  or  more  unknown  quanti- 
ties, are  indetermifiate. 

Koots  of  an  indeterminate  equation  are  often  foimd  by 
simple  inspection. 

JE.g.,  to  find  the  roots  oi  2  x  —  7  y  =  6. 

Let  X  =  0,         1,        2,      3,   4,  •  •  ■ 

then  the  corresponding  values  of  ?/ are ,  ,  ,  -,  -,  •••. 

Similarly,  find  a  set  of  roots  of.  x  +  2  y  -{-  S  z  =  10. 

Let  z  =  1 ; 

then  X  -{-2y  =  7  ; 

and  if  x  =  0,  1,  2,  3,  •  •  • 

the  corresponding  values  of  y  are  |,  3,  f ,  2,  •  •  • . 

That  is,  the  equation  is  satisfied  if 

z  =  l,  x  =  0,  y  =  l, 
or  if  z  =  1,  x  =  1,  y  =  3,  etc. 

Similarly,  we  may  start  with  2  =  2. 

211.  Sometimes  it  is  desirable  to  find  the  various  positive 
integral  roots  of  an  indeterminate  equation.  For  practical 
purposes  these  may  be  found  by  simple  inspection. 

E.g. ,  to  find  the  positive  integral  roots  of5x  +  37/=19.  Here  x  ";}>  3, 
because  if  x  >  3,  and  integral,  y  is  negative. 

-If  x  =  3,  2,  1 

then  ?/  =  a  fraction,  3,  a  fraction. 

.-.  x  =  2,  y  =  S  are  the  only  positive  integral  roots  of  the  equation. 

Graphs  and  discussion  of  equations.  For  those  who  have 
the  time,  the  study  of  the  graphic  representation  of  linear 
equations,  and  the  discussion  of  solutions  (Appendix  VIII) 
are  strongly  recommended  at  this  point. 


I 


INDETERMINATE   EQUATIONS.  203 

EXERCISES.    XOVII. 

1.  Find  three  sets  of   roots   of  each  of   the  following 
equations : 

(a)  10a;  +  32/  =  -4. 

(b)  5x-2y  =  17. 

(c)  5cc  +  232/  =  100. 

2.  Find  two  entirely  different  sets  of  roots  of  each  of 
the  following  equations : 

(a)  x-Stj-\-4:Z  =  20. 

(b)  2x-^10y-z  =  15. 

(c)  8x-7y-^5z  =  12. 

3.  Find  all  of  the  positive  integral  roots  of  each  of  the 
following  equations  : 

(a)  x  +  y  =  5. 

(b)  2^  +  102/ =  30. 

(c)  Sx  +  52/  =  20. 


4.    Find  all  of  the  positive  integral  roots  of 
x-\-2y  +  3z  =  14:. 


5.  Find  three  sets  of  roots  of  the  following  system  of 
equations : 

X  —  2y-\-4:Z  =  5. 
2x  —  y  -\-z  =  1. 

6.  Find    a    set   of    roots   of    the   following    system   of 
luations : 

2w  +  2x  +  3y-\-z  =  20. 
Sw-{-3x-\-2y  +  2z  =  25. 
4:W-{-5x  —  y  —  z  =  6. 


CHAPTER   XII. 

THE    THEORY   OF   INDICES. 

I.     THE   THREE  FUNDAMENTAL   LAWS   OF  EXPONENTS. 

212.  It  has  already  been  proved  that,  when  m  and  n  are 
positive  integers, 

1.  a"',  a"  =  a'"  +  ".  §  60 

2.  a'^-.a''^  a"*-".  §  86 

3.  (a™)"  =  «""».  §  75 

It  has  also  been  stated  (§  125)  that  a^  means  the  square 

root  of  a,  a^  means  the  cube  root  of  a,  and,  in  general,  an 

means  the  nth  root  of  a,  but  the  reason  for  this  symbolism 

has  not  yet  been  given. 

It  is  now  proposed  to  investigate   the  meaning  of  the 

negative  and  the  fractional  exponents ;  that  is,  to  find  what 

meaning  should  be  attached  to  symbols  like  3~^,  8%  16~*, 

_  1 
a"",  a    «,-••. 

We  shall  then  proceed  to  ascertain  whether  the  three 
fundamental  laws  given  above  are  true  if  m  and  n  are 
fractional,  negative,  or  both  fractional  and  negative. 

The  necessity  for  this  is  apparent.    We  know  that  a*"  • «" 

=  a"*+",  if  m  and  n  are  positive  integers,  because  a  is  taken 

first  m  times,  and  then  n  times,  as  a  factor,  and  hence 

1     ]^ 
m  -\-  n  times  in  all.     But  we  do  not  yet  know  that  an  •  am 

-+I  1      \  i_i 

=  a"    »».     Neither  do  we  know  that  a"» :  <x«  =  a'"    ",  nor  that 

a-"*  •  a-"  =  a-"—",  nor  that  a""*  •  a^  =  a   '"^n,  etc. 

204 


t 


THE   THEOllY    OF   INDICES.  205 

II.     THE   MEANING  OF   THE   NEGATIVE  INTEGRAL 
EXPONENT. 

213.  The  primitive  idea  of  power  (§  8)  was  a  product  of 
equal  factors.  The  primitive  idea  of  exponent  was  the 
number  which  showed  how  many  equal  factors  were  taken. 

According  to  this  primitive  idea  the 

3d  power  of  a  meant  aaa,  written  a^, 
2d       "         "  "         aa,       "         a^-, 

but  there  was  no  first  power  of  a,  because  that  is  not  the 
product  of  any  number  of  a's,  nor  any  zero  power,  fractional 
power,  or  negative  power. 

But  since  a^  means  aaa,  or  a^  -=-  a, 

and  a^      " 

.'.  it  is  reasonable  to  define  a^      as 

and      "  "  "        "  ^0       " 


I 


d,  in  general,  to  define 
being  a  positive  integer. 


214.    For  this  reason  we  define 

a^  to  mean  a, 

a'         "         1, 

1 

a-       "         - 

a" 

n  being  a  positive  integer. 
But  it  is  evident  that  a  ^  0. 


aa, 

a 

a""  -T-a, 

a, 

(( 

a^^a, 

1, 

a 

a  -7- a, 

1 

—  J 
a 

ii 

1  -T-a, 

1 

a 

1 

a 

1 

—  J 

206  ELEMENTS   OF  ALGEBRA. 

Illustrative  problems.     1.  Express  2"^  as  a  decimal  frac- 
tion. 

1.  2-2  i3y  definition,  means  —  • 

2.  -  =  -  =  0.125. 
23      8 

2.  Express     _       with  positive  exponents. 

-.  ah  ah  «  «„  ^ 

1.  — — -  means §  214 

a3 

2.  This  equals  ^.  §  161 

0 


3.  Express  -g^  in  the  integral  form. 
X  y 


§150 


x3y2      aj 

2.  =x-i?/.  §214 

The  expression  cc- 1?/  is  as  much  a  fraction  as  is  - ,  but  it  is  not  in 
the  form  of  a  common  fraction. 

4.  Simplify  (2-2)-2. 

1.  2-2  means  —  §214 


<«"■  •■  ©■ 


§214 


.22> 

3.  This  equals  (22)2  which  equals  2*.                §§  161,  76 

5.  Simplify  \_(2-^)-^y\ 

1.  2~i  means  -J. 

2.  (i)-i      "      2. 

3.  2-1        "      i 

4.  .-.  the  expression  reduces  to  ^. 


THE   THEORY   OF  INDICES.  207 

EXERCISES.    XCVIII. 

Express  exs.  1-4  without  exponents. 

-  ^  f- p-:' (?r-  — .-■(r 

Express  exs.  5-9  with,  positive  exponents. 

9.    2(t-3,   t^-^   (-cc)-^. 

Express  exs.  10-16  in  the  form  of  common  fractions,  with 
positive  exponents  for  the  factors. 

,  .-  ,     ..  ,  a-^^-^'     2-^3-2 

10.       -^^;;-r-     •  11. 


12-    ^ ^, .  13.    |£c-V-^H-|iC2/V. 

14.      [(£C-«)-«]-«.  15.      [(1  -  £C)-2(1  -  £C2>)-|-1_ 

16.    a-%-^cH^,  a-'^b-^'c-P. 
Simplify  exs.  17-20. 

2-8      4-5      e-7      68  ^-6      ^-a      J-c 


•    3- 

~^'l 

-4 

7-6 

•  77' 

18. 

ar''    c-^ 

&-« 

19. 

- 

-  a-^ 

■      1  +  a- 
2    3  +  «- 

-3 

3«^  +  l 

(- 

-a)- 

20. 

- 

a-2. 

(2-i)^ 

•  2- 

-«.6t2.(^,- 

•  2)- 2. 

208  -  ELEMENTS   OF   ALGEBRA. 

IIL     THE  MEANING  OF   THE  FRACTIONAL  EXPONENT. 

215.  We  have  now  found  the  meaning  of 

1.  The  positive  integral   exponent  greater  than  1,  the 
primitive  meaning  of  exponent; 

2.  The  unit  exponent ; 

3.  The  zero  exponent ; 

4.  The  negative  integral  exponent. 

216.  It  remains  to  find  the  meaning  which  should  attach 
to  the  fractional  exponent. 

The  expression  a'^  means  aaaa, 
and  if  the  exponent  is  half  as  large, 

a^  or  aa  is  the  square  root  of  a^, 
and  if  the  exponent  is  half  as  large, 

a^  or  a  is  the  square  root  of  a^. 
.'.  if  an  exponent  half  as  large  indicates  a  square  root, 

a'^  should  mean  the  square  root  of  a. 

Hence,  a^  is  defined  to  mean  the  square  root  of  a,  and, 
1 

in  general,  a'^  is  defined  to  mean  the  7ith.  root  of  a. 

217.  The  reason  for  this  is  also  seen  from  the  fact  that 
• .  •  «"*  •  «-"*  •  •  •  to  n  factors  =  a'"" . 

.'.  (/^"  •  a»  '  •  •     "  "       should  equal  a"^»'^  or  «"  or  a. 

1 
.'.  «"  should  be  defined  to  mean  the  nth  root  of  a. 

p 

218.  And  since  a*""  =  («"')«,  so  a't  should  be  defined  to  be 

r  1 

identical  with  {a'^y. 
p 
Hence,  we  define  a*i  to  mean  the  ^th  power  of  the  qth  root 
_H  p 

of  a,  and  ^    ^  to  mea?i  the  reciprocal  of  a*. 


THE    THEORY   OF    INDICES. 


209 


219.    The  following  identities  involving  fractional  expo- 
nents are  also  true  and  will  now  be  proved. 


1. 

a"Z*«c«  •  •  •  =  {(xbG  •••)«. 

Proved  in  §  220 

2. 

1                 1                    m 

§  221 

3. 

m              pm 

§  222 

4. 

11         J_           11^ 

§  224 

220.    To  prove  that  a»b»&^  ■  ■■  =  (ahc  ••■)« 


1.  Let 

1  1 

2.  .-. 

1  1 
X"  =  (a"6«)» 

Ax.  8 

3. 
4. 

1       1 
=  ab. 

§76 
§217 

5.  .-. 

x  = 

1           1  1 
(a6)«,  or  a«&»  = 

1 
(a6)«. 

Axs.  9,  1 

1  1  1 
0.  .-.  a«6»c»E 

1  1 
E  (a6)«c«  E 

1 
E  (a6c)",  and  so  or 

I  for  any 

number  of  factor^. 

Similarly, 

6" 

221.    To  prove  that  (a"*)"  =  (a^y  =  a^. 

1       112 

1.  {aaa  •  ■  •  to  m  factors)"  =  a^a^a'^  •  •  •  to  m  factors.         '    §  220 

1         1 

2.  I.e.,  (a"»)"  =  (««)"'. 

1  ^ 

3.  But  (««)»*=  a«.  Def.  §  218 

Hence,  a"  may  be  considered  either  as  the  mth  power  of 
the  nth.  root  of  a  (as  defined  in  §  218)  or  as  the  nth  root 
of  the  mth  power  of  a. 


210  ELEMENTS   OF  ALGEBRA. 

But  §  221  must  be  understood  to  apply  only  to  the  abso- 
lute values  of  the  roots. 


E.g.\ 

(42)*  =  16*  =  ±  4, 

but 

(4*)2  =  (±2)2=+4. 

222.    To 

prove 

m               pm 

that  a«  =  op^. 

1.   Let 

m, 

X  =  a». 

2.   .-. 

x»  =  a"*. 

Ax.  8  and  §  221 

3.    .-. 

^pn  ^  ^pm^ 

Ax.  8  and  §  75 

4.   .-. 

pm 

X  =  a^". 

Ax.  9  and  §  218 

5.    .-.  a''  =  a^''.  Ax.  1 

Hence,  both  terms  of  a  fractional  exponent  can  be  multi- 
plied or  divided  by  the  same  number  "without  altering  the 
value  of  the  expression. 

223.  The  student  should  understand  clearly  that  §  222  is 
true  not  because  the  exponent  is  a  fraction.  The  exponent 
is  merely  an  expression  in  the  form  of  a  fraction,  and  hence 
a  proof  like  that  of  §  150  has  no  application  to  this  case. 
The  laws  of  fractions  apply  to  fractional  exponents  only  as 
they  are  proved  to  do  so. 

11         JL  i  1 

224.  To  prove  that  («»»)»  =  a""»  =  («»)"«. 

1.  Let  a;  =  (a'»)«. 

}_ 

2.  .-.  jc"  =  a"*.  Ax.  8 

3.  .-.  x""^  =  a.  Ax.  8 

j_ 

4.  .-.  X  =  a"*".  Ax.  9 

11        j_ 

5.  .-.  («'«)«  =  «"««,  and  similarly 

1  j.        j_ 
(a«)'»  =  a'»«. 


THE   THEORY   OF   INDICES.  211 

EXERCISES.    XCrX. 

Find  the  absolute  value  of  each  of  the  expressions  in 
exs.  1-3. 

1.  4^  9^  8'%  32^,  81^. 

2.  25^  125^,  32^,  64^,  625i 

3.  16"^,  36"^,  343-«,  1331"^,  14641-?. 

Write  in  integral  form,  with  negative  or  fractional  expo- 
nents, the  expressions  in  exs.  4-9. 

1 


a  a  -{-  b  a  1 

4.    -T-  + 


V^       ^a-b       V^-\-  ^a       *' 
5.    a^b  -{-b-\^  +  VaT~b  —  ^a  —  b. 

1         1 V^  +  V^     1      1 

'    a-\U      bVb  V^  «^'      ^' 

7.  Vl  H-  a%    -v^l  --  a%    -^a^  h-  b^,  1 -- V^, 

8.  ^2  _^  (  V^  4-  Vl  -f-  a)  --  (^3  +  V'^)  --  V^. 

9.  ^ywF,  vv^^,  Vi  +  ^^Vc,  ^1  -5- (a  +  ^)'.   • 

Write  the  following  without  negative  or  fractional  expo- 
nents, using  the  old  form  of  radical  sign  (V)  and  the- 
common  fraction : 

m         m  +  l 

10.  ah^,  ah^,  ic%  £c   2  ,  x^y^,  zK 

11.  a~^,  a'h~^,  x^y~i,  ic"*-"?/"*-^". 

I 

12.    ^^ --a^-a-^^-a^ +{a-b-'^)-\ 

a"^  -r-  a~i 


212  ELEMENTS  OF  ALGEBRA. 

IV.     THE   THREE  FUNDAMENTAL   LAWS  FOR  FRACTIONAL 
AND   NEGATIVE  EXPONENTS. 

225.    Laws  1  and  2.      To  prove  that 

o m  .  on  :=:  o m  —  n 

if  m  and  n  are  fractional,  negative,  or  both  fractional  and 
negative. 

a.  Let  them  he  fractional  and  positive.     We  have  first  to 

p       r_  Pj^r 

prove  that  a'^-a^  =  a^    *. 


1. 

p         r               ps        qr 

a^'a'  =  a'^'-a^' 

§  222 

2. 

1             1 
=  {apy  •  (a'^y 

§  221 

3. 

1 
=  (aP'  ■  a^y 

§  220 

4. 

1 
=  (aP'  +  '^y 

§  60 

5. 

ps  +  qr                    p      r 

=  a    '^'    ,  OT  a''    ^ 

§  221 

3/ 


This  shows  that  a  case  like  V a^  •  Va^  can  be  easily 
handled  by  fractional  exponents,  thus : 

2  4  2,4  _22 

a^  -a^  =  a^    ^  =  a^\ 

3/ —        5/— 

To  see  that  V«.^  •  Va^  equals  the  15th  root  of  a'^^  is  not  so 
easy  by  the  help  of  the  old  symbols  alone. 

We  have  also  to  prove  that  a'> :  a^^  a*    *. 

The  proof  is  evidently  identical  with  that  just  given, 
except  that  the  sign  of  division  replaces  that  of  multipli- 
cation in  the  first  member,  and  the  sign  of  subtraction  that 
of  addition  in  the  second  member. 


THE   THEORY   OF   INDICES.  213 

b.  Let  one  exponent  he  negative  and  either  integral  or 
fractional.     We  have  then  to  prove  that 


1.  a"*  •  a- 

2. 

3. 

We  have  also  to  prove  that  <x'"  :«,""  =  a"'~<^~'*>  =  a'"+". 

The  proof  is  evidently  identical  with  that  just  given, 
except  that  the  sign  of  division  replaces  that  of  multipli- 
cation, and  the  sign  of  subtraction  that  of  addition. 


^m+(-n)^ 

or 

a"•""^ 

a"" 

§§  214,  218 

a"" 

§  156,  cor.  2 

a— «. 

§§  86,  225,  a 

c. 
fraci 

Let  both  exponents  be  negative 
ional.     We  have  then  to  prove 

a?ic?  either  integral  or 
that 

a-"'-a-"  =  ^_  „«  +  (-„) 

=  a-^ 

[  — n 

1. 

§§  214,  218 

2. 

_     1 

§  156 

3. 

1 

—  ^m  +  « 

§§60,  225,  a 

4. 

=   »-"»-". 

§§  214,  218 

As  an  illustration  of  the  value  of  these  laws,  consider 
the  case  of  — —  :  ^ 

Here  we  have 

a~^'a~^  =  a~  ^^"^^  =  a~  ^, 

or  the  20th  root  of  -,  a  result  not  so  easily  reached  by  the 
older  notation. 


214  ELEMENTS  OF  ALGEBRA. 

226.    Law  3.     To  prove  that  (a™)°  =  a™",  if  m  and  n  are 
fractional,  negative,  or  both  fractional  and  negative. 

a.  Let  m  be  fractional  or  negative  or  both,  n  being  a  posi- 
tive integer. 

1.  From  §§  60,  225,  it  follows  that 

aPa^a"-  '-•  =  aP  +  9  +  r+--- 
if  p,  q,  r,  "■  are  fractional,  negative,  or  both  fractional  and 
negative. 

2.  And  ifp  =  q  =  r  =  --'  =  m,  and  there  are  n  factors, 

then  ,     , 

(a'^y  =  a"""", 

whether  m  is  positive  or  negative,  integral  or  fractional, 
provided  %  is  a  positive  integer. 

b.  Let  m  and  n  be  positive  fractions.      We  then  have  to 

p   r  pr 

prove  that  (ai)^  =  ai^ 


1.  Let 

x  =  {a^)\ 

2.  Then 

p 

Ax.  8  and  §  221 

3. 

pr 

§  226,  a 

4.  .-. 

a;9«  =  aP\ 

Ax.  8  and  §  221 

6.  .-. 

pr 

X  =  a^\ 

Ax.  9 

6.  .-. 

pr             pr 

c.  Let  n  be  negative  and  either  integral  or  fractional,  m 
being  positive.      We  have  then  to  prove  that  (a™)"''  =  a""""^. 

1.  (a"*)-"  =  -^  §  214 

2.  -  —  §  75 

3.  =  a-"^.  §  214 


THE   THEORY  OF  INDICES.  215 

d.  Let  m  be  negative  and  either  integral  or  fractional,  n 
being  positive.      We  have  then  to  prove  that  (a~™)°  =  a~™°. 


I 


a"/ 


1.  («—)"  =  (^- j  §  214 

2.  -^  §75 

3.  =  a-^,  §  214 

e.  Let  m  and  n  &e  negative  and  either  integral  or  frac- 
tional.     We  have  then  to  prove  that  (a"~™)~"  =  a™". 

1.  ^a--r^^Q-^~^  §214 

2.  ^(^l^i.y^(^.«)n  §214 

3.  =  a^^.  §  75 

The  value  of  this  law  may  be  seen  by  the  solution  of  a 
few  problems.     Consider  for  example  the  case  of 


This  expression,  thus  written  in  the  older  style,  does  not 
strike  the  eye  as  simple  ;  but  since  1  -?-  Va^  may  be  written 
a~i,  the  expression  reduces  to  {ai)^,  which  equals  a. 
Consider  also  the  more  complicated  expression 


have 

_r      9         qr  r^—q^         qr 

X'(X      ^      '')r^-Q^  =  X-X  «'"         r^-<l^  =  X-X-^  =  X^=1. 


[To  simplify  this  without  the  assistance  of  negative  and 
fractional  exponents  would  be  more  difficult. 


216  ELEMENTS   OF   ALGEBRA. 


V.     PROBLEMS   INVOLVING  FRACTIONAL   AND   NEGATIVE 
EXPONENTS. 

227.  It  has  now  been  proved  that  we  can  operate  with 
expressions  involving  negative  or  fractional  exponents  just 
as  if  these  exponents  were  positive  integers.  Exercises 
involving  such  exponents  will  now  be  given. 

The  student  should  see  the  distinct  advantage  in  using 
the  fractional  exponent  instead  of  the  old  form  of  radical 
sign,  except  in  cases  like  the  expression  of  a  single  root, 
and  in  using  the  negative  exponent,  except  in  cases  like  the 
expression  of  a  simple  fraction.  This  has  been  shown  on 
p.  215,  but  it  is  worth  while  to  consider  the  matter  further, 
that  the  student  may  become  entirely  familiar  with  the  use 
of  the  modern  symbols. 

E.g.,  while  it  is  easier  to  write   Va  than  a%  and  -  than  a-i, 

because  we  are  more  accustomed  to  the  forms  Va  and  - ,  it  is  much 

a 
easier  to  see  that 

than  to  see  that  the  equivalent  expression 

V  (1  -  Va;2)3 
Similarly,  it  is  easier  to  recognize  in 

x's  _l_  2  x^"^  +  1  =  0 
the  quadratic  form 

x^  +  2  x^  +  1  =  0, 
than  to  recognize  it  in 

2  8/—-  2  1/ 

Vx8  +  2   V^  +1  =  0. 

It  is  doubtful  if  students  would  readily  grasp  the  significance  of 
the  form  a^  +  2  a2  va  +  a  Va  ;  but  when  written  a^  +  2  a^  +  a*  it  is 
seen  to  be  the  square  of  a^  +  a^. 


THE   THEORY   OF   INDICES. 


217 


I 

^^K  Illustrative  problems.  1.  Eemove  the  parentheses  from 
^^pB~^  H- 2/"'^)""^,  expressing  the  result  with  positive  expo- 
^Rents. 

B  (X-l  -2/-l)-2  =  X2  ^y2.  §226 

W      2.  Multiply  x-^  +  x-^  +  1  by  x-^  —  x-^  +  1. 

Since  we  can  multiply  as  if  the  exponents  were  positive,  we  have  the 
following : 

Check. 
x-2  +  x-i  +  1  3 

g-2  _  a;-i  +  1  1 

X-4  -I-  X-8  +  X-2 

-  X-3  -  X-2  -  X-1 
X-2  +  X-l  +  1 

x-4  +  x-2  +1  3 

Detached  coefficients  should  be  used  in  practice. 

3.  Divide  x'^  +  3x-'^  +  Sx-^  +  1  by  x'^  +  1. 

Since  we  can  divide  as  if  the  exponents  were  positive,  we  have  the 
following  : 

Quotient  =  x-2  +  2x-i  +  1 

x-i  +  l|x-3  +  3x-2  4-  3x-i  +  1 

X-3  +       X-2 


Check.     8  --  2  =  4. 


2x-2  +  3x-i 
2 x-2  +  2x-i 


+  1 


x-i  +  1 
Detached  coefficients  should  be  used  in  practice. 


4.  Solve  the  equation  x  ^  —  3x    o  +  2=:0. 


x2-  3x  +  2  =  (x-  2)(x  -  1), 

x~*  -  3x~*  +  2  =  (x-*  -  2)(x-^  - 

(x-^  -2)(x-3  _  1)  =0. 

x~5  =  2,  or  x~^  =  1. 

x-i  =  23  =  8,  or  x-i  =  V^  =  1. 


8-1 


or  X  =  1,  a::d  these  roots  check. 


218 


ELEMENTS   OF  ALGEBRA. 


EXERCISES,    C. 
Eemove  the  parentheses  and  simplify  in  exs.  1-8. 

1.  [(-arf. 

2.  (ar'  +  y-y. 

1    _1_ 

4.      [(«'»  +  '*)"'-".(«»')«]'«*. 
_i  _i 

6.     \l(x-Y^T^~%   («^^^^)*. 


7.    (203-2- 2/- 2)-^    V64[(x-2/)-6]i 

Express  with  positive  integral  or  fractional  exponents,  in 
simplest  form,  exs.  9-14. 

9.    -y/a-^b^p. 
11.    ■V^a2'"Z»3m2_ 

_^  3/ — 

13.    a   ib-'^Wcd^ 


10.    V^v^- 

m  -(-  re  / 

12.     v«y-< 


14.     ^a-"'b-^'"c-"'\ 

Perform  the  multiplications  indicated  in  exs.  15-19. 

15.  3a~^-4:a~^'2a^;    ax—^y-'^-bx^'y'^. 

16.  a^-a^;    3a-^-^c^-4.a-%^c-^;    a;-^.(-aj^). 

17.  5-y/x^-2xhf',    -a-'b^c^d-''-a'b-^c'd-\ 

18.  (a;2  +  2xy  +  y^)-  (x-'  -  2  x-hf-"-  +  y-'). 

19.  (a;-3  +  3  x-'y  +  3  ic-y  +  y^)  ■  (x-'  +  2x-hy  +  y^). 


THE   THEORY   OF  INDICES. 

Perform  the  divisions  indicated  in  exs.  20-30. 

21.  4:abh^:2b^ck 

22.  (ah^  —  ah^  +  4  Jb^)  :  ah\ 

23.  4a;-* +  11^"' -45  by  2ic-i-3. 

24.  a-"  — a-^  +  1  by  a-^  —  a'^  +  l. 

25.  (4  x^yl  -  9  xhjl)  :  (2  xJy^  +  3  aj^i). 

26.  x~^  +  2  a;-2^-i  —  3  y-^  by  cc-^  —  y-\ 

27.  3  a-4  :  5  a't,  x^  :  J  [x-i^/'i (ccy)^]-^^-^. 

28.  16cc-8  +  6a3-2  +  5a;-i-6  by  2ic-i-l. 


219 


29.    V^4  x-h/z^  :  [(1 :  ^12a;V^«')  •  '^lOSx-h/z-^y 


30.    ic-^  — 2ic-* 


4ic-«  +  19x-2_3l£c-i  +  15  by  a;" 


3 1 .    rind  the  remainder  when  a;-  ^  — 11  cc"  ^  + 1 0  is  divided 
byx-i-1. 


32.    Also  when  x~^  +  (a  —  3)  x' 
divided  by  x~'^  —  3. 


(b-3a)x-^-3bis 


33.  Factor   2  x-^  —  9  x-^  —  S  x-'^  -\- 15,    negative   expo- 
nents being  allowed  in  the  factors. 

34.  Also  6a;-«  +  x-2-5a;-i-2. 

35.  Also  6a;-^  +  17ic-2-18a:-i-45. 

36.  Also  1  —  9a7^  —  486 x^,  fractional    exponents  being 
allowed  in  the  factors. 


220  ELEMENTS   OF  ALGEBRA. 

VI.     IRRATIONAL   NUMBERS.     SURDS. 

228.  Rational  and  irrational  algebraic  expressions  have 
already  been  defined  (§  98).  But  in  algebra  it  is  often 
necessary  to  use  numbers  which  are  irrational. 

229.  A  rational  number  is  a  number  expressible  as  the 
quotient  of  two  integers. 

E.g.,  3  =  f ,  0.666- ••  =  |,  f 

230.  An  irrational  number  is  a  number  which  is  not 
rational. 


E.g.,  2'  or  V2,  (1  +  2^)^  or  V^l  +  V2,  V-  1. 

231.  Irrational    numbers  which    are  not  even    roots  of 
negative  numbers  are  often  called  surds,  but  in  elementary 
works  the  term  is  still  further  limited  to  irrational  roots  of j 
rational  numbers,  or  to  such  roots  combined  with  rational 
numbers. 

E.g.,  V2  and  3  +  Vs  are  the  types  here  treated,  but  not  v  2  +  Vs  j 
and  V—  5. 

232.  Surds  are  classified  as  follows : 

1.  According  to  the  root  index,  as 

quadratic,  or  of  the  second  order,  as  Vs, 


cubic,             "        "    third         " 

"    ^7, 

quartic,  or  biquadratic, 

"    V^, 

quintic. 

«    ^, 

sextic 

"   V^, 

and  in  general  as 

n-tic,  n  being  a  positive  integer. 

"   -v^. 

THE   THEORY   OF   INDICES.  221 

2.  Similar  or  dissimilar  (if  they  have  a  single  term), 
according  as  the  surd  factors  are  or  are  not  the  same. 

E.g.,  2V3,  4V3,   -  7  V3  are  similar  surds. 
2  V3,  3  V2  are  dissimilar  surds. 
V2  .  V3,  5  V3  are  similar  as  to  V3  but  dissimilar  as  to  V2. 

3.  Pure  or  mixed  (if  they  have  a  single  term),  according 
as  they  do  not  or  do  contain  either  real  factors  or  dissimilar 
surd  factors. 

E.g.,  V3  is  a  pure  surd,  but  2  V3  and  V5  •  V3  are  mixed  surds. 

4.  According  to  the  number  of  terms  in  the  expression 
when  simplified,  as 

monomial  surds,  as  'v2,  3V2, 

binomial       "       "   V2  +  Vs,  5  +  V2, 

trinomial      "       "  2  +  Vs  +  -y/j, 

and,  in  general,  polynomial  surds. 

5.  According  to  simplicity.  A  surd  is  said  to  be  in  its 
simplest  form  when  all  the  factors  that  are  perfect  roots 
are  expressed  without  the  root  sign,  when  the  index  is  as 
small  as  possible,  and  there  are  no  fractions  under  the 
radical  sign. 

E.g.,  V9,  Vi,  V^,  Va^x,  are  not  in  the  simplest  form.     For 
V9  =  3, 

Vi  =  Vvl  =  V2, 

VJ  =  V|  =  VfT2  =  ^  V2. 
Va^  =  aVx. 

The  fractional  exponent  is,  in  general,  more  convenient  in 
all  operations  involving  surds.  The  two  forms  of  the  radical 
symbol  are  used  here  in  order  that  both  may  be  familiar. 


222 


ELEMENTS   OF  ALGEBRA. 


233.  Convention  as  to  signs.  When  we  consider  an  ex- 
pression like  V4  + V9  we  see  that  it  reduces  to  (±2)+(±3), 

and  hence  to 

+  2  +  3  =  5, 

+  2  -  3  =  -  1, 

-2  +  3  =  1, 

_  2  -  3  =  -  5. 

But  for  simplicity  it  is  agreed  among  mathematicians 
that  in  expressions  of  this  kind  only  the  absolute  values  oj 
the  roots  shall  be  considered  unless  the  contrary  is  stated. 

Hence,  Vi  +  Vg  =  2  +  3  =  5 ,  but  ±  Vi  ±  ^9  =  5,  -  1,  1,  or 
-  5.     (Compare  §  192.) 


EXERCISES.    CI. 

1.  Classify  according  to  the  index  of  the  root : 

(a)   ■^.         (b)   ^.  (c)  a*.  (d)  x^. 

2.  Classify  as  similar  or  dissimilar : 
(a)  2V2,  5V2,  8 -21        (b)  2v^,  -V^,  i-^. 

3.  Select  the  surds  from  the  following : 
(a)    V2.  (b)  4*.  (c)  VV2  +  V3. 

4.  Classify  as  pure  or  mixed : 
(a)   V47.         (b)  3-v/5.         (c)  ab^.         (d)   V2-V^J 

5.  Classify  according  to  the  number  of  terms : 
(a)  ah^.  (b)    V2  +  -^.  (c)  2  +  V3  +  -^J 

6.  rind  the  value  of  each  of  these  expressions : 

(a)   V4  +  V9  +  VI6.       (b)  V^  + V25  +  -^  +  ^/32. 
(c)   -v^l728  -  Viii  +  Vi69  -  13. 


THE   THEORY  OF  INDICES.  223 

234.  Reduction  of  surds.     It  has  been  shown  (§  217)  that 

a  =  (a")".     Hence,  it  follows  that  a  number  can  be  reduced 
to  the  form  of  a  surd  of  any  order. 

E.g.,  2  =  Vs,  the  form  of  a  surd  of  the  3d  order. 
Similarly,  V2  can  be  reduced  to  the  form  of  a  surd  of  the  5th  order, 
for  2'  =  (2^)^  or  V^,  or  ^V^. 

Similarly,  Vi  =   Vis  =    Vie,  a  surd  of  the  10th  order. 

Hence,  mixed  surds  can  ahvays  be  reduced  to  pure  surds. 
E.g.,  :■  aVb=V^, 

3V5  =  y/S^-6  =  Vl35. 

235.  Since  it  is  desirable  to  have  the  number  under  the 
radical  sign  as  small  an  integer  as  possible,  it  is  often 
necessary  to  reduce  surds  to  their  simplest  forms  (§  232,  5). 

Vl36=  V33.5  =3  Vs. 

\18         \32.2       \32.22      6 
Hence,  in  the  case  of  fractions  under  the  radical  sign  we 
multiply  both  terms  by  the  smallest  number  which  will  make 
pAe  denominator  the  required  power,  then  extract  the  indi- 
cated root  of  the  denominator,  and  reduce  the  remaining 
surd  as  much  as  possible. 


E.g.,  ^/i^^      f±:^^lV39. 

^'  \13       \132  13 


236.  Since  in  multiplying  surds  it  is  desirable  to  have 
them  of  the  same  order,  it  is  often  necessary  to  reduce 
several  surds  to  equivalent  surds  of  the  same  order,  the 
order  always  being  as  low 'as  possible. 

E.g.,  \^  •  V3  =  2^  •  3's  =  2^  •  3^  =  (23 .  32)*  =  Vs^  =  V72. 


224  ELEMENTS  OF  ALGEBRA. 

EXERCISES.    CII. 

1.  Reduce  the  following  numbers  to  the  forms  of  surds 
of  the  orders  indicated  : 

(a)  5,      3d  order.  (b)  2,      6th  order. 

(c)  i,  4th  «  (d)  10,  5th  « 
(e)  11,  2d  "  (f)  12,  3d  " 
(g)  -2,  2d  "  (h)  -5,3d  " 
(i)  3,      5th    ^'  (j)  -2,  6th     « 

2.  Eeduce  the  following  to  pure  surds  : 

(a)  2V3.  (b)  3V2.  (c)  2V^. 

(d)  5-2*.  (e)  ah^c.  (f)  a^2^\ 
(g)  3  V2.  V3-  Vs.             (h)  ah^c. 

3.  Eeduce  the  following  numbers  to  the  forms  of  surds 
of  the  orders  indicated  : 

(a)    ^abc%  9th  order.  (b)  V'a^,  14th  order. 

(c)   a/5,     30th     "  (d)  3*     .15th     « 

(e)  5*,        20th      "  (f)  10*     15th      " 

(g)    v'4,       8th      "  (h)  V'S,    60th      " 

4.  Eeduce  the  following  to  equivalent  surds  of  the  same 
order,  the  order  being  as  low  as  possible  in  each  case  : 

(a)  V^,   Vb.  (b)  V3,    ^,    a/2. 

(c)  2^,  3^  4^.  (d)  v^,   V3,   A^. 

(e)  ah^,  ah^.  (f)  ^,   V^,    V^. 

(g)  7^,  9^,  111  (h)  2,   V2,   v^,   -v^,   -^5. 


THE   THEORY   OF   INDICES. 


226 


237.  Addition  and  subtraction  of  surds.  Irrational  expres- 
sions may  evidently  be  added  and  subtracted  the  same  as 
rational  expressions,  by  taking  advantage  of  some  con- 
venient miit. 

Check. 

E.g.,  aVx+  hVx  —  c-\^        1 

3/-  n/- 

-  c^x  +  cVz         0 

gVx  4- 6Vx  +  cVz        3 

2aVx  + (26- c)Vx +  cVz        4 

Similarly,  required  the  sum  of  V24,  V54.  and  —  V96.  Here  we 
have,  each  surd  being  reduced  to  its  simplest  form, 

V24=       V4T6    =      2V6 

V54=       V9~^    =      3V6 

-  V96  =  -  V16.6  =  -4V6 

Hence,  the  sum  is  v6 

Similarly,  required  the  sum  of  Vs,  V27,  —  2  V2,  and  Vis.     Here 

we  have  2V2  +  sVs  -  2V2  +  4V3  =  7 Vs. 

In  general,  however,  the  sums  of  surds  can  only  be  indi- 
cated as  Vs  +  ^,  -  -v^  +  "V^. 


I 


EXERCISES,    cm. 
Simplify  the  following : 

1.  V72  +  VIO8  -  V32  -  V243. 

2.  ■>^+ v^375- -v^elS  +  lOv^. 

3.  V^  +  ^^/a^^  -  ^a^^  ■  V&. 

4.  (a^b)^  -  a^^b  +  a%^c. 

5.  Vl47  +  V243  -  V363  +  V432  -  V507. 

6.  -v^l715  4-  -V^3645  +  ^6655  +  V^640  -  39  -^5. 


7.    Vx^  -{-  5x^  +  6x^  -  4:x  -  S  +  -Vx^ -  4.x^ -{-6x^  —  ^x  +  1. 


226  ELEMENTS   OF  ALGEBRA. 

238.  Multiplication  of  surds.  In  general,  products  involv- 
ing irrational  numbers  must  be  indicated,  as  3  V2,  or 
expressed  approximately,  as  3  V2  =  3  •  1.414  •  •  •  =  4.24  •  •  •. 

E.g.,  3  V2.2  Vs  =  3  V8-2  Vo  §236 

=  6V72.  §  220 

This  result,  while  it  still  leaves  a  root  to  be  extracted 
and  a  multiplication  to  be  performed,  is  more  compact  than 
the  indicated  product  3  V2  •  2  Vs. 

Similarly,  to  square  3  V2  +  2  Vs. 

(3a^  +  2V3)2  =  (3V2)2  +  2(3V2)(2V3)  +  (2V3)2     §69,  1 
=  18 +  I2V72+4  V9. 

It'  is  understood  that  no  results  are  to  be  expressed 
approximately,  in  decimal  form,  unless  so  stated. 


EXERCISES.    CIV. 

Perform  the  following  multiplications : 
1.    3V|.2-^.  2.    V2--^.v^.V3. 

3.    (3-5V3)2.  4.    V^.^.^^.-4^. 

5.    V7  .  -^7  •  ^.  6.    {-^/a-b  +  Va  +  bf. 

7.    2V2.3V3.5V6.  8.    VT2I.  V^.-J/14641. 

9.    (2  +  8  V3)(4-5V3). 

10.  3  V2.2^3.4V^.5V^. 

11.  V^/a  +  V^ .  V^Va  -  V^. 

12.  (V2  + V3)(2  V2-5  V3). 


13.    5  ■y/{a  +  2  ^»)2 .  3  V(a  +  2  bf. 


THE    THEORY   OF   INDICES.  227 

239.  Division  of  surds.  To  divide  an  irrational  number 
hy  a  rational  number  is  equivalent  to  multiplying  by  the 
reciprocal  of  the  rational  number,  and  hence  it  may  be  con- 
sidered as  a  case  of  multiplication. 

E.g.,  IS  merely  -  •  (a  +  V6),  or  -  +  -  V6. 

240.  Division  by  a  surd  usually  reduces,  without  much 
difficulty,  to  division  by  a  rational  number,  as  shown  in  the 
following  example : 

To  divide  V2  +  Vs  by  Vs,  we  have : 

Vi  +  Vs  _  V5(V2  +  V3) 

assuming  that  we  can  multiply  both  terms  of  the  fraction  by  VE  with- 
out changing  the  value,  as  we  can  in  the  case  of  rational  multipliers 
(§  150).     This  equals 

,  or  i(^10  +  ^1^)- 

6 

241.  In  the  preceding  example  we  have  reduced  the 
•fraction  to  an  equivalent  fraction  with  a  rational  denomi- 
[nator.      The  process  of  rendering  a  quantity  rational  is 

called  rationalization. 

The  advantage  of  rationalizing  the  denominator  is  seen  by  consid- 
^ering  the  computation  necessary  to  find  the  approximate  value  of 

V2  +  Vs 

•     Here  there  are  three  square  roots  to  be  extracted,  fol- 

V5 
lowed  by  one  addition  and  by  one  division  with  a  long  divisor. 

But  in  the  case  of  ^  ( VlO  +  Vl5)  there  are  only  two  square  roots 
to  be  extracted,  followed  by  one  addition  and  by  one  division  with  a 
!  short  divisor. 

242.  The  factor  by  which  an  expression  is  multiplied  to 
produce  a  rational  expression  is  called  a  rationalizing  factor. 

E.g.,  Vs  can  be  rationalized  by  multiplying  it  by  V2. 


228  ELEMENTS   OF   ALGEBRA. 

243.  Since  the  prohleyn  of  division  by  surds  reduces  to 
that   of  the   rationalization   of  the   divisor,    exercises    in 
rationalization  will  'first  be  considered- 
Illustrative  problems.     1.  By  what  expression  may  a-b^ 

be  multiplied  in  order  that  the  product  shall  be  rational. 

1.  •.•  X'*  •  X       «  =  X, 

2.  .-.  a%^ ■  a}-%^-^  =  ah. 

3.  .-.  a^~V~^  or  a^h^^  is  a  rationalizing  factor.  There  are  evi- 
dently any  number  of  rationalizing  factors,  since  we  may  multiply  this 
one  by  any  rational  expression.     This  is,  however,  the  simplest  one. 

2.  By  what  expression  may  V a"*  •  V5^  be  multiplied  in 
order  that  the  product  shall  be  rational  ? 

1.  V#.  W  =  o^lfi  =  ah^h. 

2.  Evidently  a^h^h  •  a}'~%^~^  will  equal  a^^,  a  rational  expression. 

3.  .-.  a^h^  is  a  rationalizing  factor. 

3.  By  what  expression  may  a  -f-  V^  be  multiplied  in 
order  that  the  product  shall  be  rational  ? 

1.  •••  (X  -  2/)  (X  +  y)  =  x2  -  y'^,  §  69 

2.  .'.         (a  —  \h)  {a  +  Vo)  =  a^  —  b,  sl  rational  expression. 

3.  .-.  a  —  Vft  is  a  rationalizing  factor. 

244.  And,  in  general,  the  conjugate  of  a  binomial  quad- 
ratic surd  (§  69,  3)  is  a  rationalizing  factor  of  that  surd. 

4.  Find  a  rationalizing  factor  for  'y/a  ±  VZ  ±  Vc. 
\.  :•  {x  +  y  +  z)  {-  X  +  y  +  z)  {X  -  ij  -\-  z)  {x  -^  y  -  z) 

=  2  X-^?/2  +  2  2/222  +  2  22x2  _  X*  -  2/4  -  2*, 

2.  .-.  any  trinomial  quadratic  surd  of  the  form  v a  ±  \b  ±  vc 
can  be  rationalized  by  multiplying  it  by  the  product  of  the  other 
three  trinomials.     E.g.,  the  rationalizing  factor  for  V2  —  VS  +  Vs  is 

(^^  +  Vs  +  V5)  (-  V2  -f-  V3  +  V5)  ( V2  +  V3  -  Vs). 


THE   THEORY  OF   INDICES. 


229 


EXERCISES.    CV. 

Find  the  simplest  rationalizing  factor  for  each  of   the 
following  expressions : 


1.  ahh^(h. 

3.  (^h^c^. 

5.  2  + Vs. 

7.  3-V2. 

9.  V5-I. 

11.  aJn'^c  »  . 

13.  Vt  +  Vs. 


2.  V7-V5. 

4.  Va  —  V^. 

6.  Va  +  ^»  +  c. 

8.  V5-V2-V3. 

10.  V5  +  V7  +  Vil. 

12.  V2  +  V7-Vli. 


14.     Vi*  +  6  +  Va  —  6. 
Illustrative  problems  in  division.     1.  Divide  Vl2  by  V3. 

§220 


2.  Divide  V5  by  -^2. 

V5_      V53 
■v/2  V22 


=Ai 


§220 


6  h"  ■  53 

\-2^ 


=  i  V2000. 


3.  Divide  V2  +  V3  by  V2  -  VS.     That  is,  rationalize 

V2  +  V3 
ithe  denominator  of  the  fraction  — 7= 7=- 

V2- V3 

1.  The  rationalizing  factor  for  the  denominator  is  evidently 

V2  +  Vs. 

2_   (V2  +  V3)(V2  +  V3)^2  +  2V6  +  3^      ^^   ^  ^  ^/^^^ 
(V2  +  V3)(\^  -  V3) 


2-3 


230  ELEMENTS  OF  ALGEBRA. 

EXERCISES.    CVI. 

Perform  the  divisions  indicated  in  exs.  1-16. 
1.    6:4-v'24.  2.    24  :  (2 Vt  -  6). 

3.    ^a%^'.2^^h.  4.    15V24:3V^. 

5.    58:(8+V35).  6.    12  ^192  :  4  ■V^729. 

7.    16-V^^^':8V'^«^.  8.    90  :  (5  V3  -  VSO). 

9.    10Vl2:2  Vl8:4V8. 

10.  -^(a^  -2  6)^:  V(a«  -  2  h). 

11.  (VT2-  Vl8+  V6):  V2. 

12.  (3V5-8  V2):(3  V3-4V5). 

13.  (18  -  16  V5)  :  (4  -  VS  -  2  V3). 

14.  (7  Vi2  -  4  V27)  :  (8  V3  +  2  V2). 

15.  (15  V8  +  10  Vt  -  8  V2  -f  5)  :  -  4  Vs. 

16.  (3  V3  -  2  V2)  :  (5  VS  -  3  V2  -  2  VS). 

Rationalize   the  denominators   of  the  fractions  in  exs. 
17-23. 

17.    i4±4-  18.  ^' 


11^  -  5^  2-3^  +  5^ 

7  +  3V7  3_5i_2^ 

19. p=-  20. -• 

12-6  Vll  3  +  5i  +  2^ 


21. 


2                                                      2m 
22. 


(^2  4-  If  +  («;2  _  ^^t  (^^  _^  ^)*  +  (a  -  ^) - 

1 

23. 


a;(l  -a2)i_2/(l  4_a2^i 


THE   THEORY   OF   INDICES. 


231 


245.  Roots  of  surds.  The  roots  of  perfect  powers  of 
surd  expressions  can  often  be  found  by  inspection  or 
extracted  in  the  ordinary  way. 

1.  To  find  the  square  root  of  a  +  4  -\fab  +  4  J. 


1.  ••• 

2.  .-. 

Check. 


V/2  ^  2fn  +  n^  =  ±  (/  +  n) , 
V9  =  ±3. 


2.  To  find  the  fifth  root  of  the  perfect  fifth  power 

a2  -  5  a%^  +  10  a/'h^  -  10  ab  +  ^  ahl^  -  h^. 

This  is  readily  seen  to  be  a*  —  6^.  §  82 

To  check,  let  a  =  6  =  1.     Then  0^  =  0.     If,  however,  we  wish  to 

jheck  the  exponents,  let  a  equal  any  square  and  6  equal  any  cube. 

?.gr.,  leta  =  9,  6  =  8.     Then 

(3  -  2)5  z=  243  -  810  +  1080  -  720  +  240  -  32. 

3.  To  find  the  square  root  of  7  +  4  Vs. 

1.  If  this  can  be  brought  into  the  form/2  _j_  2/n  -|-  71^,  the  root  will 
in  the  form  ±  (/  +  w).  §  69 

2.  We  first  make  the  coefficient  of  the  second  term  2,  because  of 
le  2/n,  and  have  7  +  2  Vl2. 

3.  And  ',•  12  is  the  product  of  3  and  4,  and  7  is  the  sum  of  3  and 
I,  we  have 

V7  +  4V3  =  V4  +  2  V3T4  +  3  =:  ±  ( Vi  +  V3)  =  ±  (2  +  V3). 
Check.     Square  2  +  Vs. 

4.  To  find  the  square  root  of  8  —  2  Vl5. 

1.  As  in  ex.  3  we  attempt  to  bring  tjhis  into  the  form/2  ^  2/n  +  v?. 

2.  •.•  15  is  the  product  of  5  and  3,  and  8  is  their  sum,  we  have 

V8-2  Vl5  =  Vs  -  2  Vl5  +  3  =  ±  ( V5  -  V3). 

Of  these  results,  only  the  positive  one  is  usually  considered  in 
)ractice. 

Check.     Square  Vs  —  V3. 


232 


ELEMENTS  OF  ALGEBRA. 


EXERCISES.  CVII. 

1.    Extract  the  square  roots  of 

(a)  a-2-yf2ab^2h.  (b)  a-2a'-^oJ'. 

(c)  3  a  -  8  VS^  +  16.  (d)  a*  -  2  a^&3  4.  ^,1 . 


(e)  26t-  V200a  +  25.  (f)  x^^^x'^Hj- 

2.  Extract  the  cube  roots  of 

(a)  8  -  12  Va  +  6  tt  -  a  V^. 

(b)  a-3V^^2^3Z>V^-Z»l 

(c)  a;^  -  3ic2  v^  +  3£c  Vy  -  V^. 

(d)  cc^  Vx  —  3  x^  V?/  +  3  a;  V^^y  —  y. 

3.  Extract  the  fifth  roots  of 

(a)  1  -  52/^  +  10  y"  -  l^if  +  St/^  -  y\ 

(b)  32-80 -^  +  80-v^2/^-407/v^^  +  10y/V^-2/2 

4.  Extract  the  square  roots  of 

(a)  8-2  Vt.  (b)   I  +  V2. 

(c)  8  +  VeO.  (d)  9  -  4  V2. 

(e)  10  -  V96.  (f )  I  +  ^  V6. 

(g)  IOV7  +  32.  (h)  II2  +  40V3. 

5.  Extract  the  square  roots  of 


(a)  2x4-2V^ 


(b)  2  a;  +  2  Va;2  _  1. 


(c)  ah  —  2a  ^ ah  —  a'^. 

(d)  x^  -{-  X  -{-  y  -\-  2  X  Vic  +  y. 


^i  ff 


THE   THEORY   OF   INDICES. 


233 


VII.     THE   BINOMIAL  THEOREM. 

246.    It  has  been  shown  (§  80,  the  proof  being  given  in 
Appendix  I)  that  if  n  is  a  positive  integer 


(a  +  by  =  a"  +  na'^-^  +  -^—^ «"~ 


2^2 


^.(.-l)(»-2)^,_3^3^ 


It  was  proved  by  Sir  Isaac  Newton  that  this  is  true  even 
if  w.  is  negative  or  fractional.  The  proof  is,  however,  too 
:  difficult  for  the  student  at  this  time. 

Assuming  that  the  binomial  theorem  is  true  whether  ?i 
positive  or  negative,  integral  or  fractional,  it  offers  a 
raluable  exercise  in  the  use  of  negative  and  fractional 
jxponents. 

E.g.,  •.•  (a  +  6)« 

,,     n(n  —  l)       „,-,    n(n  —  l)(n  —  2) 
2  2  •  o 

•.  y/a+b={a-\-h)^ 

2  Z  •  o 


=  a^+ia-h  -ia-^62 

+tV«-^6« 

.-.  V5=(4  +  l)^ 

=4^  +  1.4-^-1.4-* 

+  T^6-4-^ 

=  2  +i           -J^ 

+  5l^ 

_/i   1  ^\—a. 

(l+X)3-(''^^ 

=  1  +(-3)x+^^i^ 

-V,-3(-3-l)(-3-2)^,  1 

=  1  -3x        +6x2 

-10x3                              + 

234  ELEMENTS   OF  ALGEBRA. 

EXERCISES.    CVIII. 

1.  Expand  to  four  terms  (1  +  x)~^. 

2.  Also  l/Vl-£c. 

3.  Also  Vil  =  V16-2  =  4  (1  -  1)^. 

4.  Find  the  5tli  term  in  the  expansion  of  (1  —  x)~^. 

5.  Also  in  the  expansion  of  (1  +  xy. 

6.  Also  in  the  expansion  of  (1  —  x)i. 

7.  Find  VTo   by  expanding  (9  +  1)^   to   four    terms, 
reducing  these  to  decimal  fractions  and  adding. 

8.  Similarly  for  V82  =  (81  +  1)^. 

9.  Similarly  for  -^  =  (27  +  1)^ 
10.    Similarly  for  V37  =  (36  +  1)K 

REVIEW  EXERCISES.    CIX. 

1.  Divide  x^  —  4:X^a^  +  Qx^a^  —  4:X^a^  +  a'^  by  x^  — 
2  x^a^  +  ak 

2.  Simplify  3  (a^  +  x^y  -  4  (a"^  +  x^)  (a*  -  x^)  +  (a^  - 
2x^y. 

3.  Simplify  -3 


4.    By  inspection  find  the  square  root  of 

(a)  4  a-^  +  4  4-  a^. 

(b)  at-2af +  5al-4ai  +  4. 

(c)  cc  +  ?/  +  ^  rf  2  x^y^  —  2  a;'^^^  —  2  ?/^^^. 

(d)  a^  +  4  aV  +  10  aifi  +  12  ahy  +  9  ?/t. 


THE   THEORY   OF   INDICES. 

5.    Simplify  (3*  +  3^  +  3^  +  1)  (3^  -  1). 


235 


6.  Factor  36  x^  —  65  cc^  —  36,  fractional  exponents  being 
lUowed  in  the  factors. 

7.  Also  4  a;^  —  4  x^y^  +  9  y^- 

8.  Solve  the  equation  x'^  -\-  ^  x'^  -\-  2  =  0. 

9.  Also  4  x^  -  15  a;*  +  14  =  0. 

10.  Also  £c^  —  5  cc^  +  6  =  0. 

11.  Extract  the  square  root  of 

12.  Also  of  25ic-*-30cc-^3/  +  49ic-y-24a;-y  +  162/*. 

13.  Extract  the  cube  root  of 
aj-6  _  9  ^-5  _^  33  ^-4  _  g3  ^-3  ^  gg ^-2  _  3g  ^-i  _^  3 

14.  Also  of 
8ic2  4.  48a;t  +  60cc"3  -  80a;  -  90 cc"^  +  108  x^  -  27. 

15.  Also  of 

8  a*  +  48  a^b  +  60  a^b^  -  80  a^Z^^  _  90  a^b"^  +  108  ai^*^  -  27  ^»«. 

16.  Also  of 
■^^x-i  +  ^\ic-i  +  |a7~t  +  7  a;-T%  +  3  a;-i  +  |x-tV  + 1. 

17.  If  a^  =  b%  show  that  (t  V  =  «^"^- 

\^      /V3-- 

V3-V2y        VV3  +  V2y 

19.    Simplify 


,  V3  +  V2\2  ,    /  V3  -  V2V 
18.    Simplify  (      ,-^ — 7=  )   +  (  -:= 7=  ]  • 


CHAPTER   XIII. 

COMPLEX   NUMBERS. 

I.     DEFINITIONS. 

247.  Certain  steps  in  the  growth  of  the  number  system 
have  already  been  set  forth  in  §  24,  but  are  here  repeated 
for  reasons  which  will  be  obvious. 

1.  The  positive  integer  suffices  for  the  solution  of  the 
equation  a;  —  3  =  0,  since  x  =  3  satisfies  the    . 

0  12  3 

equation.     We  can  represent  such  a  number 

by  a  line  three  units  long,  as  in  the  annexed  figure,  the  unit 

being  of  any  convenient  length. 

2.  The  positive  fraction.  If,  however,  we  attempt  to 
solve  the  equation  3  x  —  2  =  0,  either  we  must  say  that 
the  solution  is  impossible  or  we  must  extend  the  idea  of 
number  to  include  the  positive  fraction.  Then  ic  =  f  sat- 
isfies the  equation.  We  can  represent  such  a  number  by 
dividing  a  line  one  unit  long  into  three  parts  and  taking 
two  of  them. 

3.  The  surd.  If  we  attempt  to  solve  the  equation 
ic^  —  2  =  0,  either  we  must  say  that  the  solu- 
tion is  impossible  or  we  must  extend  the  idea 
of  number  to  include  the  surd.  Then  V2 
satisfies  the  equation.  We  can  represent 
V2  by  the  diagonal  of  a  square  whose  side 
is  one  unit  long.  This  is  evident  because  the  square  on 
the  hypotenuse  equals  the  sum  of  the  squares  on  the  two 
sides  of  the  right-angled  triangle. 

236 


COMPLEX   NUMBERS.  237 

4.  The  negative  number.  If  we  attempt  to  solve  the 
equation  a;  +  2  =  0,  either  we  must  say  that  the  solution 
is  impossible  or  we  must  extend  the  idea  of  niunber  to 
;  include  the  negative  number.  Then  x  =  —  2  satisfies  the 
equation.  We  can  represent  such  a  number  by  supposing 
^tlie  negative  sign  to  denote  direction,  a  direction  opposite 

that  which  we  assume  for  positive  numbers. 

248.  The  numbers  thus  far  described  in  this  chapter  are 
jailed  real  numbers. 

249.  The  imaginary  number.  If  we  attempt  to  solve  the 
jquation  x''^  -{- 1  =  0,  either  we  must  say  that  the  solution 
is  impossible  or  we  must  extend  the  idea  of  number  still 
further. 


The  equation 

a;2  +  1  =  0 

eads  to 

x^^  =  -l, 

jvhich  leads  to 

x  =  ±V 

1, 

'which  cannot  be  a  positive  or  a  negative  integer,  fraction, 
or  surd  (§  126). 

250.    We  call  an   even   root  of   a  negative  number  an 
imaginary  number. 

The  term  "  imaginary  "  is  unfortunate,  since  these  num- 
bers are  no  more  imaginary  than  are  fractions  or  negative 
lumbers.     We  cannot  imagine  looking  out  of  a  window 
[—  2  times  or  -^  of  a  time  any  more  than  V—  1  times.     The 
:"  imaginary  "  is  merely  another  step  in  the  number  system. 
?he  name  is,  however,  so   generally  used   that  it  should 
jontinue  to  designate  this  new  form  of  number. 

To  the  ancients,  negative  numbers  were  as  "  imaginary  "  as  V—  1 
to  us.    It  was  only  when  some  one  drew  a  picture  of  V2  (see  §  247,  3), 
[of  —  1,  and  later  of  V^l,  that  these  were  uuderstood. 


238 


ELEMENTS   OF   ALGEBRA. 


251.    As  with  fractions,  surds,  and  negative  numbers,  it 
is  necessary  to  represent  the  imaginary  graphically  by  a 

line,  or  in  some  other 
concrete  way,  in  order 
to  make  its  nature  clear 
to  the  beginner.  * 

In  this  figure  the 
multiplication  of  +  1 
by  —  1  swings  the  line 
OA^  through  180°  to 
the  position  OA^. 

As  a  matter  of  custom 
this   line    is    supposed   to 
swing  as  indicated  by  the 
arrows,  opposite  to  the  movement  of  clock-hands,  counter-clockwise. 


' 

' 

B. 

■i-aYn 

Ba 

A. 

/ 

A^3 

+iV=:i 

V. 

B, 

—2 

0    ^; 

-f  2 

B. 

.-2TP, 

y 

r 

252.  That  is,  since  ( V—  1)^  means  V—  1  •  V—  1  or  —  1, 
the  multiplication  of  +  1  by  V—  1  •  V—  1  swings  -f  1 
through  180° ;  therefore  the  multiplication  of  +  1  by 
V—  1  should  be  regarded  as  swinging  it  through  half  of 
this  angle,  or  90°,  to  the  position  0^2- 

Or  we  may  say  that  since  multiplication  by  V—  1  twice, 
carries  OA  through  180°,  therefore  multiplication  by  V—  1 
once  should  carry  it  through  90°. 

Similarly,  —  1  multiplied  by  V—  1  •  V—  1,  or  —  1  mul- 
tiplied by  —  1,  swings  OA^  the  rest  of  the  way  around  to 
0^1 ;  hence,  —  1  multiplied  by  V—  1  should  be  looked 
upon  as  swinging  it  to  the  position  OA^. 

253.  Hence,  we  represent  + 1 V— 1  (or  -f-  V^),  +2  V^, 
+3  V  —  1,  .  • .,  by  integers  on  the  perpendicular  OY,  upward 
from  0,  and  -iV^  {or  -V^),  -2V-i,  -3V^,  •••, 
hy  integers  on  the  negative  side  of  this  line,  i.e.,  on  OF', 
downward  from  0. 


COMPLEX   NUMBERS.  239 

254.    Hence,  it  appears  that  the  symbols  +  V—  1   and 
—  V—  1  wre,  like  -f  and  —,  symbols  of  quality  and  may  be 

I  looked  upon  as  indicating  direction. 
E.g.^  +3  indicates  3  units  to  the  right, 

-3  "  "         "      left, 

+  3V^         "  "      up, 

V  —  3  V  —  1         "  "      down. 

li 


255.    Since   Va6  =  Va-  -\/b,   we  say    that   V—  3  shall 

[ual  V3-1  =  V3 .  V-  1.     Hence, 

Every  imaginary  number  can  be  written  in  the  form 
a  V --  1,  where  a  is  real,  though  possibly  a  surd  or  a  frac- 
tion, and   v  —  1  is  the  imaginary  unit. 


E.g.,  to  represent  3  V—  1,  we  measure  3  units  upward  from  the  0 
point  on  the  line  X'X ;  to  represent  —  V—  2,  we  reduce  this  to  the 
form  —  V2  •  V—  1,  then  construct  a  line  equal  to  Vi,  as  in  §  247,  3, 
and  lay  this  off  on  0Y\ 


EXERCISES,    ex. 

Solve  the  following  equations,  expressing  the  results  in 
the  form  a  V—  1. 

1.  x2  =  -9.  2.  3cc2_^2  =  0. 

3.  5a;2  =  _5.  4.  a;2V2  =  -3. 

5.  ^2_^5  =  0.  6.  5x^  =  -125. 

7.  a;2  +  4  =  0.  8.  cc2  +  20  =  -5. 

Kepresent  graphically  the  following  imaginary  numbers : 

9.  V^.  10.  V^.  11.  -5V^. 

12.    V-32.  13.    3V^.  14.     V2.V^. 


15.    -  V-  16.  16.    2V-9.  17.    -^V-12. 


or 

{x-2f 

or        {dc, 

-2  +  V 

^(^-2- 

whence 

or 

240  ELEMENTS   OF   ALGEBRA. 

256.  The  complex  number.  If  we  attempt  to  solve  the 
equation  ic^  —  4cc  +  5  =  0by  factoring,  we  may  write  it  in 
theform  ^^  _  4^  +  4  -  (- 1)  =  0, 

-(-!)==  0, 

V-1)=0, 

x  =  2-  V^Ti^ 
x  =  2+  V^. 

Hence,  it  appears  that  each  root  is  the  algebraic  sum  of 
a  real  number  and  an  imaginary. 

Such  a  number  is  said  to  be  complex. 

257.  As  with  positive  and  negative  integers,  fractions, 
surds,  and  imaginaries,  we  proceed  to  make  the  nature  of 
the  complex  number  more  clear  by  resorting  to  a  graphic 
representation. 

If  we  wish  to  represent  the  sum  of  2  and  —  3,  we  pass 

" ';:::::"_::__;,  from  zero  2  units  to  the  right 

-10       12  and  then  3  units  to  the  left, 

and  we  say  that  the  sum  is  the  distance  from  0  to  the  point 
where  we  stop. 

The  fact  that  the  absolute  value  of  the  sum  is  less  than  the  sum  of 
the  absolute  values  of  the  addends  is  no  longer  strange  to  us,  because 
we  have  become  accustomed  to  this  in  dealing  with  negative  numbers. 

258.  Similarly,  to  represent  the  sum  of  3  and  2  V—  1  we 
pass  from  zero  3  units  to  the 
right  and  then  2  units  upward 
(for  2  V—  1)  and  we  say,  as  be- 
fore, that  the  sum  is  the  distance 
from  0  to  the  point  where  we  stop. 

The  fact  that  the  absolute  value  of  the  sum  is  less  than  the  sum  of 
the  absolute  values  of  the  addends  is  no  more  strange  than  it  is  in  the 
case  of  2  4-  (—  3). 


COMPLEX   NUMBERS. 


241 


EXERCISES.    CXI. 

Represent  graphically  the  following  complex  numbers : 
1.4+  V^^.  2.    5  -  2  V^. 

3.    5  +  2V^.  4.    -^_V:r^. 

5.    -5-2  V^.  .6.    -  3  -  3  V^^. 


7.    -i  +  iV3-V-l. 


8. 


i-iV3.V-l. 


259.  Symbolism  of  complex  numbers.  Instead  of  writing 
the  symbol  V—  1,  the  letter  i  is  usually  employed. 

This  letter,  standing  for  imaginary,  seems  to  have  been  first  used  in 
this  sense  by  Euler  in  1777.    , 

Then  V^  =  2  V^  =  2  i, 

V^  =  i  V3,  etc. 


Llso, 

i'  =  -l, 

P  =.-!.{  =  -  i, 

i'=(:ir={-ir  =  i, 

i^  =l.iz=  i, 

i^  =  i.i  =  i^  =  -l, 

{^  =  —  1 .  i  =  —  i, 

i^  =  -i.i  =  -({y  =  -(-l)=l; 

,  in  general, 

^^«  =  1, 

i'-  +  ^  =  i, 

{'-  +  '  =  -1, 

•4n  +  3  ^  _  .• 

I'"  '  "  = 


EXERCISES.    CXII. 

Represent  graphically  the  following  complex  numbers  : 
1.    2  +  3i.  2.    4  +  2i.  3.    i^  +  ^i. 

4.    i^  +  i\  5.    i*  +  t^-  6.    i^-h2i\ 


242 


ELEMENTS   OF  ALGEBRA. 


II.     OPERATIONS   WITH   COMPLEX   NUMBERS. 

260.    Complex  numbers  are  subject  to  all  of  the  laws  of 
rational  numbers   and   the   operations   do  not  materially 

differ  from  those  already 
familiar  to  the  student. 

Illustrative  problems.    1. 

Represent  graphically  the 
sum  of  2  +  3  ^  and  —  3  —  t. 

Starting  from  0  we  lay  off 
+  2  (to  the  right),  then  Si 
(upward),  OA  being  2  +  3  i. 
From  A  we  then  lay  off  —  3 
(to  the  left),  then  —  i  (one 
unit  downward),  reaching  B. 
Then  the  sum  is  OB,  the  distance  from  0  to  the  point  where  we  stop. 

2.  Add  1,  —  Y  +  i *  V3,  and  —  ^  —  ^i  Vs ;  then  repre- 
sent the  sum  graphically. 

1 

-  i  -ji  Vs 
Sum  =      0 

Graphically,  we  lay  off  1  from 
0  to  A.  From  A  we  lay  off  —  ^, 
then,  iiVi" (i.e.,  i-M.73.--, 
or  0.87  i),  reaching  B.  From  B  we  lay  off  —  ^,  then  —  i  i  Vs,  reach- 
ing O.     Hence,  the  sum  is  zero. 


Y 

B 

Av 

0 

A 

3. 

Multiply  2  +  3  ^ 

2  +  3i 

3  -2t 

:  by  3  - 

21. 

6  +  9i 

=  6  +  9i 

= 

6  +  9i 

-  4  i  -  6  i2 

=     -4ti 

-6(- 

-1)  = 

6-4i 

12  +  5i 
Simply  multiply  by  i  as  if  it  were  any  other  letter,  but  in  finally 
simplifying  remember  that  i^  =  —  1. 


COMPLEX   NUMBERS.  243 

4.  Divide  12  +  5i  by  3-2i. 

Multiply  both  terms  of  the  fraction 
12  +  5i 
3-2i 
by  the  conjugate  of  the  denominator.     Then 

(3  +  2  i)  (12  +  50  _    26  +  39  i    _  26  +  39  i  _ 

(3  +  2i)(3-2i)  ~9-4(-l)  ~        13       ~     "^     *' 

5.  Cube  —  1  +  i  *  Vs. 

•••  (/  +  n)8  =P  +  SPn  +  3M  +  n\ 
.-.  (-i  +  iiV3)3 

=  -i  +  3.i-iiV3+3.(-i).f.(-l)  +  f(-l).iiV3 

=  -i  +  |iV3  +  f-fiV3 

=  1. 
Hence,  —  i  +  i  i  V3  is  a  cube  root  of  1. 


6.  Extract  the  square  root  of  —  16  +  30  *. 

•.•  a  +  2  V^  +  6  =  [±(Va  + V6)]2,  §245 

id       •.•  -  16  +  30  i  can  be  written  9  +  2  V  -  9  •  25  +  ( -  25), 


I: 
.-.  -  16  +  30i  =  9  +  2  V-  9  •  25  +  (-  25) 
=  [±(3  +  V325)]2 
=  [±(3  +  5i)P. 
."•  ±  (3  +  5  i)  is  the  required  square  root. 
The  solution  is  seen  to  consist  simply  of  making  the  coefficient  of 
the  square  root  2,  and  then  separating  —  16  into  two  parts  whose  prod- 
uct is  -  225.     (See  §  245,  3.) 

The  addition  (including  subtraction)  of  complex  numbers 
has  been  represented  graphically.  It  is  also  possible  to 
represent  the  other  operations  graphically,  but  the  expla- 

I nation  is  too  difficult  for  an  elementary  text-book. 
I 


7.  Extract  the  square  root  of  a^  +  2  abi  —  Z»^. 

This  is  evidently  the  same  as  a^  +  2  ahi  +  (bi)^. 
Hence,  the  square  root  is  ±  (a  +  hi). 


244  ELEMENTS  OE  ALGEBRA. 

EXERCISES.    CXIII. 

1.  Find  the  following  sums  and  represent  each  solution 
graphically. 

(a)  5-7  i  and  5  +  7  i.  (b)   -  2  -  3  ^  and  2  +  3  i. 

(c)   1,  -  1,  i,  and  -  i.  (d)   -  6  +  2i  and  6  +  2  i 

(e)  1,^  +  ii-^,  -i  +  iiV3,   -1,  -i-|'iV3,  and 
i-i^V3. 

2.  Multiply 

(a)  3  -  4^-  by  5  +  21.  (b)   -  ^  +  ^ ^  by  i  +  ^ /. 

(c)  2  +  9t  by  9  +  2/.  (d)   -4  +  2^  by  -4-2/. 

(e)   -^  +  i/V3  by  -  i--|^V3. 

3.  Divide 

(a)  10  by  3  -  i.  (b)  4  +  22  i  by  7  +  i. 

(c)  1  +  8  i  by  2  +  i.  (d)  1  +  8  i  by  2  +  3^. 
(e)  7  +  61  i  by  4  +  7  i.  (f )  3  +  6  i  by  3-6  i. 

4.  Eaise  the  following  to  the  powers  indicated  : 

(a)  p\  (b)  (2  +  3  ly.     (c)  (-  i  -  i  ^^sy. 

(d)  (2  +  .•)^.  (e)   (3  -  5  .)^        (f )   (-  i  +  i  V:=l)3. 
(g)   («  +  My.       (h)   (2  -  7  i)^        (i)   (-  i  -  i  V33)3. 

5.  Extract  the  square  root  of 

(a)  3  +  4^'.  (b)  5 +  121.  (c)  -5-121 

(d)   _45-28^.  (e)  24  -  10  i.  (f)  15  -  8  i. 

(g)  -\^-  +  21  i.  (h)   -  i  -  ^^.        (i)   -  f  +  i. 


COMPLEX  NUMBERS.  245 

REVIEW   EXERCISES.    CXIV. 

1.  Simplify  the  expression 
-  i/[10  +  2  V5  4-  ( VS  +  1)  i]. 

2.  Also  the  expression  (Vs  -{- iy / i (— 1  -\-  V—  3)^ 

3.  Also  the  expression 
1  + V^Y  A  + A'  /-I- v^Y 

2     A  V2;  V     2     ;• 

4.  By  factoring,  solve  the  equation  8a?^  —  35a^  +  12  =  0. 

5.  By  the  Remainder  Theorem  determine  whether  x  —  i 
is  a  factor  of  x^  -\-  5x^  -\-  4. 

6.  Find  the  times  between  4  and  5  o'clock  at  which  the 
hands  of  a  watch  are  at  right  angles. 

7.  By  factoring,  find  four  different  roots  of  the  equation 
1  =  0.     (Two  are  imaginary.)     Check. 


I 

I^P     9.    Find  to  two  decimal  places  the  values  of  x  and  y  in 
the  following : 

"  =7.935. 


8.    By  substituting  the  three  numbers 

1,   -i  +  i^V3,   -^-^iVs, 
)r  X,  show  that  they  are  the  roots  of  the  equation  ic^  —  1  =  0. 


^     ,     y    _ 


3.579  '   5.793 
^     +  -I—  =  5.397. 


1 


9.753      7.539 

10.    The  sum  of  two  numbers  is  16,  and  the  sum  of  their 
reciprocals  is   double   the  difference   of   their  reciprocals, 
hat  are  the  numbers  ? 


CHAPTER   XIV. 

QUADRATIC   EQUATIONS   INVOLVING   ONE 
UNKNOWN   QUANTITY. 

I.    METHODS   OF  SOLVING. 

261.  A  quadratic  equation  (or  equation  of  the  second 
degree)  involving  one  unknown  quantity  is  an  equation 
which  can  be  reduced  to  the  form  ax^  -^hx  -\-  c  =  ^,  a,  b,  c 
being  known  quantities  and  a  not  being  zero. 

E.g.,  3ic2  +  2ic  +  3  =  0, 

a;2  +  1  =  0, 

■i-a^2_^ic  V2  =  0, 

are  quadratic  equations  involving  one  unknown  quantity. 
So  is  the  equation 

2x^  +  Zx^-5x  +  l  ={2x^  +  l){x-  1), 

because  it  can  be  reduced  to  the  form  ax^  -\-  bx  +  c  =  0. 
Similarly  for 

although,  in  general,  multiplication  by  any  f{x)  is  liable  to 
introduce  an  extraneous  root  (§  185). 

But  0  •  a;2  +  4  (c  -  5  =  0 

is  not  a  quadratic  equation ;   neither  is 

2ic8  +  ic  +  l  =x^  +  x''  +  ^x, 

nor  x''  +  x  +  l={x  +  l){x- 1). 

246 


QUADBATIC  EQUATIONS. 


247 


The  equation  x^  -\-  x^  -\-  4  =  0 

is  not  a  quadratic  equation  in  x,  but  it  is  one  in  x^,  for  it 


the  same  as 


So 


(xy  +  (x^)  +  4  =  0. 


+-+2=0 


X'        x 

I,  without  reduction,  a  quadratic  equation  in  ->  or  x~^,  and 
(^a  +  xy  -\-2(a  +  x^  +  3  =  0 
a  quadratic  equation  in  a.  +  x%  and 

x^  -\-x  +  3  Vic2_|_^  ^  4 


a  quadratic  equation  in  Vic^  +  cc. 

262.  The  quadratic  equation  ax"^  -^  bx  -^  c  =  0  is  said  to 
be  complete  when  neither  b  nor  c  is  zero ;  otherwise  to  be 
incomplete. 

The  coefficient  a  cannot  be  zero,  because  the  equation  is  to  be  a  quad- 
ratic (§  261). 

jB.flr.,  x2  +  2  X  —  3  =  0  is  a  complete  quadratic  equation, 

It  x2  _  3  ^  0 

x2  +  2  X  =  0  are  incomplete. 

Older  English  works  speak  of  an  equation  of  the  form 

ax^  +  c  =  0  as  a  pure  quadratic, 
id  ax^  +  So:;  +  c  =  0  as  an  affected  quadratic. 

The  following  are  further  examples  of  complete  (aifected) 
[uadratic  equations : 

(x  -  1)^  +  (^  _  l)i  +  5  =  0,  in  (x  -  1)*; 

11  ir- 

-—  +  —=  + 7  =  0,  in  V^; 
£c*  +  2  ic^o  +  1  =  0,  in  x^^. 


248  ELEMENTS  OF  ALGEBRA. 

263.    Solution  by  factoring,     (a)  The  type 

(ax  +  b)  (ex  +  d)  =  0. 

One  of  the  best  methods  of  solving  the  ordinary  quadratic 
equation  is  by  factoring,  as  already  shown  in  §  123. 

Illustrative  problems.     1.   Solve  the  equation 
a^2  +  16  X  +  63  =  0. 

1.  This  reduces  to  (x  +  9)  (x  +  7)  =  0.  §  119 

2.  This  is  satisfied  if  either  factor  is  zero,  the  other  remaining  finite 
(§  123).     Hence,  either 

X  +  9  =  0,  or  X  +  7  =  0. 

3.  .-.  X  =  —  9,  or  X  =  —  7. 

Check.     Substituting  these  values  in  the  original  equation  (§  189), 
81  -  144  +  63  =  0, 
49  -  112  +  63  =  0. 

2.  Solve  the  equation  2x^  =  1. 

1.  This  reduces  to  x^  =  |.  Ax.  6 

2.  .-.  X  =  ±  V|  =  ±  i  Vii.  Ax.  9,  §  235 
That  is,  it  is  not  worth  while  to  factor  as  in  ex.  1.     But  the  problem 

can  be  so  solved ;  for 

x2  -  I  =  0. 

•••  («  -  ^)  («  +  ^)  =  0- 

,  ...  X  =:  ±  V|  =  ±  i  Vli. 

Check.     Substituting  in  the  original  equation, 
2.^.14  =  7. 

3.  Solve  the  equation  Qx^  —  lx  +  2  =  0. 

1.   This  reduces  to  (2  x  -  1)  (3  x  -  2)  =  0.  §  120 

2..-.  2x-l  =0,  or  3x-2  =  0.  §123 

3.    .-.  2x=  1,  or  3x  =  2, 

and  X  =  i,  or  x  =  f . 

Check.  I  _  I  +  2  =  0,         f  -  Y  +  2  =  0. 


QUADRATIC   EQUATIONS. 


249 


EXERCISES.     CXV. 

Solve  the  equations : 
1.    x^  =  X. 


2.    X- 


7  -6cc. 


3.  ^-i  =  6. 

X^         X 

5.  9a;2_l  =  0. 

7.  ic2_j_i7^^0. 

9.  x''-2x-lb  =  (). 

11.  cc2  +  5x-14  =  0. 

13.  x^  +  19x  +  lS  =  0. 

15.  a;2-12a;-85  =  0. 

17.  x''-22x-\-121  =0. 

19.  ic2  _  24  x  + 143=^0. 


1        1 

6.  x^  =  2{12~^x). 

8.  8ic-a;2_12  =  0. 

10.  X  (10  +  £c)  =  -  21. 

12.  6a;24-7ic  +  2  =  0. 

14.  ic2  +  26  X -:  -  120. 

16.  £c  (4  —  ic)  +  77  ==  0. 

18.  3cc2-10x  +  3  =  0. 

20.  10:z;2  +  29cc  =  -10. 


264.    (b)  The  type  (x  +  a)  (x  -  a)  =  0. 
It  frequently  happens  that  it  is  easier  to  arrange  the  first 
lember  as  the  difference  of  two  squares  than  to  factor  in 
the  form  suggested  on  p.  248,  especially  when  the  numbers 
re  such  that  the  linear  factors  involve  surds. 

E.g. ,  to  solve  the  equation  x^  +  4  x  +  1  =  0.     Here  x^  +  4  x  are  the 
rst  two  terms  of  a  square,  x^  +  4  x  +  4.    The  equation  may  be  written 

x2 +  4x4-4-3=0, 

(X  +  2)2  -3  =  0, 

(X  +  2  +  V3)  (X  +  2  -  V3)  =  0, 

ice  we  are  not  confined  to  the  domain  of  rationality  (§  107)  in  our 
)lutions. 

.-.  X  +  2  +  V3  =  0,  or  X  +  2  -  V3  =  0, 

id  X  =  -  2  -  V3,  or  X  =  -  2  +  V3. 

Check.     4±4V3  +  3-8t4V3  +  1=0. 


250 


ELEMENTS   OF   ALGEBRA. 


265.  The  addition  of  an  absolute  term  to  two  terms  so 
that  the  trinomial  shall  be  a  square  is  called  completing  the 
square. 

E.g.^  to  complete  the  square  of  x^  4.  2  x  we  must  add  1 ;  to  complete 
the  square  oix"^  -{■  x  we  must  add  ^. 

266.  Since  {x  -{-  ay  =  x'^  -\- 2  ax  -{-  a'^,  it  is  seen  that  the 
quantity  which  must  be  added  to  x^  +  2  ax  to  complete  the 

square  is  the  square  of  half  the  coeffi- 
cient of  X. 

E.g.^  to  complete  the  square  for  cc2_j_8x, 
add  16,  £c2  +  8  X  +  16  being  (x  +  4)2.  To  com- 
plete the  square  f or  x  +  6  Vx  with  respect  to 
Vx,  add  9,  X  +  6  Vx  +  9  being  ( Vx  +  3)2. 

From  the  annexed  figure  it  is  readily  seen 
that  if  we  have  x2  +  cix  +  ax,  or  x2  +  2  ax,  the 
square  on  x  +  a  will  be  completed  by  adding  a2  in  the  corner. 


a          ax 

a'  i 

X            x== 

X 

ax 

a 

EXERCISES.    CXVI. 

Complete  the  squares  in  exs.  1-16. 


1       2 

1-    -i  +  -- 

£C^         X 

3.  cc  —  V^. 

5.  x'^  +  ^x. 

7.  x'^  —  lx. 

9.  4.x'' -{-^x. 

11.  x^-lOOx. 

13.  9ic2  4.36x. 

15.  100x2  +  20cc. 


2.  ^+2^. 

a^  a 


4.  x'^-Qx. 

6.  x^  +  30x. 

8.  x''-Mx. 

10.  cc2  +  10  X. 

12.  x''  —  2x',   x'^  +  ^x. 

14.  (x-iy  +  4.{x-l), 

16.  {x  +  ay  +  2(x  +  a). 


17.    In  general,  to  complete  the  square  for  x'^  +px  what 
must  be  added  ? 


QUADRATIC   EQUATIONS. 


251 


Illustrative  problems.     1.  Solve  the  equation 

1.  Completing  the  square  for  x^  +  3  x,  the  equation  may  be  written 

x2  +  3x  + 1-^  =  0. 

2.  .-.  (x  + 1)2-^  =  0. 

3.  .-.  (aj  +  |+i)(x  +  f-i)  =  0, 

(X  +  2)  (X  +  1)  =  0. 

4.  .-.  X  =  —  2,  or  —  1. 
Check.     4-6  +  2  =  0,  1-3  +  2=0. 


2.  Solve  the  equation  x  —  Va;  +  1  =  0. 


1.                                                      x=V^ 

+  1. 

Ax.  3 

2.    .-.         x2  =  X  +  1,  or  x2  -  X  -  1  =  0. 

Axs.  8,  3 

3.    .-.                            x2-x  +  ^-f  =  0. 

4.    ...(a;-i+iVg)(x-i-iV5)  =  0. 

5.    .-.                                                   »  =  i  ± 

iVs. 

Check.     i±|V5-V3±iV5 

=  i±^V5-iV6±2V5 

'=^±|V5-iVl±2V5  +  5 

§245 

=  |±iV5-|(l±  V5)  =  0. 

EXERCISES.    CXVII. 

Solve  the  equations : 


X      25 


3. 


3.    a;2-2x  =  -2. 
5.    cc2  _  9  ^  _  1  ^  Q^ 

T,.  x''-7x  +  5  =  0. 


(-9-- 


2.  1/ 

4.  x''  +  6x  +  2  =  0. 

6.  x2-6ic  +  2  =  0. 

8.  a^2_^io^_|_5^o. 


9.    a;2_pi0ic  +  25  =  0. 


252  ELEMENTS  OF  ALGEBRA. 

267.  Solution  by  making  the  first  member  a  square.  The 
method  of  §  264  may  be  modified  by  making  the  first 
member  the  square  of  a  binomial  of  the  form  x  +  a. 

E.g.,  to  solve  the  equation  cc^  +  4  cc  +  1  =  0. 
The  first  member  would  be  a  square  if  the  1  were  4,  i.e., 
if  3  were  added.     Hence,  adding  3  to  both  members, 

1.  ic2  +  4  ic  +  4  =  3.  Ax.  2 

2.  .-.  (x  +  2y  =  3, 

3.  .-.  03  +  2  =  ±V3,  Ax.  9 

4.  .-.  x  =  -2±-sfZ. 

Check.     (-  2  ±  Vsf  +  4  (-  2  ±  Vs)  +  1 

=  4T4V3  +  3-8i4V3  +  l=0. 

268.  It  therefore  appears  that  the  equation  x^  +  px  +  q  =  0 
can  be  solved  by 

1.  Subtracting  q  from  each  member  ;  then 

2.  Completing  the  square,  by  adding  the  square  of  half 
the  coefficient  of  x  (§  266)  to  each  member ;  and  then 

3.  Extracting  the  square  root  of  each  member  and  solv- 
ing the  simple  equations  which  are  thus  obtained. 

The  ±  sign  in  step  3  of  the  above  solution  is  placed  only 
in  the  second  member,  because  no  new  values  of  x  would 
result  if  it  were  placed  in  both  members. 

Suppose  it  were  placed  in  both  members.     Then 

±  (x  -I-  2)  =  ±  Vs  ;  that  is 

(1)  +  (x  +  2)  =  -}-  V3,  whence  x  =  -  2  +  V3, 

(2)  +(x  +  2)  =  ~V3,         "       x  =  -2-V3, 

(3)  -  (X  +  2)  =  +  V3,         "  -  X  =      2  +  v'S  and  .-.  X  =  -  2  -  V3, 

(4)  -(x  +  2)=-V3,         "-x=      2-V3     "     .•.x  =  -2  +  V3. 
That  is,  X  =  —  2  ±  V3,  as  in  step  4  of  the  solution. 


QUADKATIC    EQUATIONS. 


263 


Illustrative  problems.     1.  Solve  the  equation 

x^  -{-  X  +  1  =  0. 

1.  x^  +  x=  -1. 

2.  X2  +  X  +  i  =  -  1  +  i  =  -  f . 

3.  x  +  i=±iiV3. 

4.  .-.  x=  -iiiiVa. 
Check.  (-  i  T  ii  V3)  +  (-  i  ±  ii  V3)  =  -  1. 


Ax.  3 
Ax.  3 
Ax.  9 
Ax.  3 


2.  Solve  the  equation  x^  +  3x+  Vx^  +  3a;  +  7-23  =  0. 

1.  This  may  be  written  in  quadratic  form,  thus, 

x2  +  3  X  +  7  +  Vx2  4-  3  X  +  7  -  30  =  0, 
quadratic  in  Vx^  +  3  x  +  7.     This  quantity  may  now  be  represented 
jy  y,  for  simpUcity,  and 

2.  2/2  +  2/  -  30  ==  0. 

3.  .-.  y''-hy^l  =  H^- 

4.  .-.  y  +  h=±  ¥• 

6.    .-.  y=-l±  -u-  =  5,  or  -  6. 

6.  .-.  Vx2  +  3  X  +  7  =  5,  or  -  6. 

This  evidently  gives  rise  to  two  quadratic  equations  in  x.     First 
consider  the  case  ot  y  =  6. 

7.  Then  x2  +  3x  +  7  =  25. 

8.  .-.  x2  +  3x-18  =  0. 

9.  .-.  (X  +  6)  (X  -  3)  =  0,  and  X  =  -  6,  or  S, 
results  which  easily  check.  ^ 

If  y  =  —  6,  we  have 

10.  x2  +  3  X  +  7  =  36, 

11.  whence  x^  +  3  x  +  f  =  ^f ^. 

12.  .-.  x+ f  =  ±f  V5,  andx=  -  I  ±1  Vs. 
This  pair  of  results  checks,  provided  we  remember  that 

Vx2  +  3  X  +  7  =  5  or  -  G. 
For,  substituting  5  and  -  6  for  Vx^  +  3x  +  7,  we  have 
18  +  5-23  =  0, 
29  -  6  -  23  =  0. 


254  ELEMENTS  OF  ALGEBRA. 

3.  Solve  the  equation  2  a;'^  —  2  cc  =  5. 

1.  x^-x  =  ^.  Ax.  7 

2.  a;2  -  X  +  I-  =  -V--  Ax.  2 

3.  X  -  J  =  ±  i  VlT.  Ax.  9 

4.  x  =  i(l'±VlT).  Ax.  2 

Check.     (6  ±  VlT)  -  (1  ±  vTT)  =  5. 

It  is  often  possible,  in  cases  of  this  kind,  to  avoid  fractions  by  the 
exercise  of  a  little  forethought.     This  equation  may  be  written 
r.  4x2-4x  =  10. 

2'.    .-.  (2  x)2  -  2  (2  x)  +  1  =  11,  a  quadratic  in  2  x. 

3'.    .-.  .       2  X  -  1  =  ±  Vll. 

4'.    .-.  2  X  =  1  ±  VlT. 

6'.    .-.  x  =  i(l  ±  Vll). 

EXERCISES.    CXVIII. 

Solve  the  equations : 

1.   x^-^x  =  l.  2.  6X  +  4.0  —  x^  =  0. 

3.    a;2  +  8a;  =  65.  4.  x^  +  ^x --%»-  =  0. 

5.  x^  +  0.9  x  =  8.5.  6.  2.5£c2-4fa;  =  304. 
7.    3ix^-4.x  =  96.  8.  ic2_,_i32a;  =  _l33l. 
9.   x''  +  6x  +  25  =  0.  10.  7ic2-5x- 150  =  0. 

11.    4x2_5^_pg2  =  0.  12.    4..05x^-7.2x  =  U76. 

13.  (a;  +  a)2-f-2(£c  +  a)+l  =  0. 

14.  (  a;  +  -y_3|  ^  +  -  1  +  2  =  0. 


(.  +  ^y-3(x  +  l)  +  2 


15.  (ic2  +  2a;)2_3(ic2  +  2ic)  +  2  =  0. 

16.  (x^-^x-iy  +  4:(x^  +  x-l)  +  4.  =  0. 
17.    (cc  +  4)  (12  ic  -  5)  +  4^  =  (7ic2  -  10)8  -  12.75a;, 


QUADRATIC   EQUATIONS. 


255 


269.    Solution  by  formula.     Every  quadratic  equation  can 
ll)e  reduced  to  the  form  ax^  -\-hx  -\-  c  =  ^  (§  261). 

This  equation  can  be  solved  by  any  of  the  methods  already 
suggested  and  it  will  be  found  that 


h^h^ 


4taG. 


Hence,  the  roots  of  any  quadratic  equation  which  has 
)en  reduced  to  the  form  ax^  -j-  bx  +  c  =  0  can  be  written 
'down  at  sight. 

E.g.,  the  roots  of 


6x2  _  I3x  +  6  =  0  are —  ±  —  V(- 13)2 

2-6       2-6     ^         ' 

=  {i±^  Vl69  -  144 

Similarly,  the  roots  of 
2       3 


4.6-6 


+  1  =  0  are  -  = 

X  X 


2-2      2-2     ^       ' 


42.1 


=  f  ±  i  V9  -  8 
=  f  ±  i  =  1  or  i. 
X  =  1  or  2. 


270.    In  particular,  the  roots  of 

x^  +px  -\-  q  =  0  are  x  =  —  ^  ±^  ■\Jp^  —  4:q. 

E.g.,  the  roots  of  x2  +  x  +  1  =  0  are  -  i  ±  ^  Vl  -  4 

=  -i±iiV3. 


271.    The  formulas 
X  =  - 


^±J-Vb^ 

2a      2a 


4ac, 


X  = 


__P 


±iVp^-4q, 


are  so  important  that  they  should  be  Tnemorized  and  freely 
used  in  the  solution  of  such  quadratic  equations  as  are  not 
:readily  solved  by  factoring. 


256  ELEMENTS   OF  ALGEBRA. 


EXERCISES.     CXIX. 

Write  out,  at  sight,  the  roots  of  equations  1-30,  and  then 
simplify  the  results. 

1.  a;2-3a;  +  l  =  0.  2.  ic^  +  6ic  +  2  =  0. 

3.  cc^  +  dcc  — 4  =  0.  4:.  x^  —  5x-\-l  =0. 

5.  x^  +  2x  +  2  =  0.  6.  x2  +  2cc-24  =  0. . 

7.  ic2-2ic  +  3  =  0.  8.  x''-5x-36  =  0. 

9.  ic^  +  2  ic  -  3  =  0.  10.  0-2  +  7  cc  -  44  =  0. 

11.  x^-5x-S6  =  0.  12.  ^2  + 10 a^ +  5  =  0. 

13.  x^  +  7x-\-l()  =  0.  14.  x^  — 4.x -12  =  0. 

15.  12ic2  +  aj-6  =  0.  16.  x''^  4.x -45  =  0. 

17.  cc2-7ic  +  12  =  0.  18.  x2-3cc-28  =  0. 

19.  3x^-2x-\-l  =  0.  20.  ;:c2- 16a; +  60  =  0. 

21.  4:X^-\-5x  +  6  =  0.  22.  ic2  _^  10 a:  +  21  =  0. 

23.  2cc2  +  3x  +  l  =  0.  24.  6cc2-37cc  +  6  =  0. 

25.  ic2-2.l£c-l  =  0.  26.  6x^-\-5x-56  =  0. 

27.  i»2-llic-60  =  0.  28.  a;2  _^  0.6  ic  +  0.3  =  0. 

29.  a;2  -  10  a;  +  16  =  0.  30.  r^^  +  0.7  a-  +  0.1  =  0. 

31.  What  are  the  roots  of  the  equation  ax^  -j-bx  -\-c  =  0, 
iiP  =  4ac? 

32.  Show  that  if  b'^  —  4  ae  is  negative  the  two  roots  are 
complex. 

33.  Show  that  if  S^  —  4  ac  is  positive  the  two  roots  are 
real. 

34.  Show  that  if  Z'^  —  4  ac  is  a  perfect  square  the  two 
roots  are  rational. 


QUADRATIC  EQUATIONS. 


257 


272.    Summary  of  methods  of   solving  a  quadratic  equa- 
ftion.      From  the  preceding    discussion  it  appears  that  a 
luadratic  equation  is  solved  by  forming  from  it  two  simple 
ruations  whose  roots  are  those  of  the  quadratic. 

E.g.,  to  solve  the  quadratic  equation 

re  may  write  it  in  the  form 

(x  +  3)  (^  +  4)  =  0, 

rhence   cc  +  3  =  0,    or  x  +  4  =  0,    two    simple    equations 
fhose  roots,  —  3,  —  4,  are  those  of  the  quadratic. 

Or  we  may  write  it  in  the  form 

rhence  [(x  +  |)  +  i]  [(x  +  J)  -  ^]  =  0, 

md  therefore  x  +  |  +  ^  =  0, 


0, 


fwo  simple  equations  whose  roots,  —  3, 
[quadratic. 

Or  we  may  write  it  in  the  form 

x'  +  ix-\-{iy  =  {\f, 

f  whence  cc  +  7  —  i 


4,  are  those  of  the 


4,  are  those  of  the 


two  simple  equations  ivhose  roots,  —  . 
\quadratic. 

Or  we  may  simply  write  out  the  results  from  a  formula 
[obtained  by  one  of  the  above  methods. 

For  expressions  easily  factored  the  first  method  is  the 
llDest;  otherwise  it  is  usually  better  to  use  the  formula  at 
(once. 


258  ELEMENTS   OF  ALGEBRA. 

Illustrative  problems.  1.  Solve  the  equation 
a;  +  3  X  +  1  _Sx  —  5  3a;  —  3 
x-\-5~x  +  3~3x  —  7~3x  —  5 

The  denominators  are  such  as  to  suggest  adding  the  fractions  in 
each  member  separately  before  clearing  of  fractions.     Then 

* = i 

(a;  +  3)(x  +  5)       (3x-5)(3x-7) 

2.  Multiplying  by  i  (x  +  3)  (x  +  5)  (3  x  -  5)  (3  x  -  7), 

(3x-5)(3x-7)=(x  +  3)(x+_5).  Ax.  6 

3.  .-.  8  x2  -  44  x  +  20  =  0,  (Why  ?) 
or                           2x2-llx  +  5  =  0. 

4.  This  is  easily  factored  (§  263),  and 

(X  -  5)  (2  X  -  1)  =  0. 
6.    .-.  X  =  5  or  i. 

Check.     For  x 


2.    Solve  the  equation -\ -\ =  0. 

X  —  1      X  —  2      X  —  3 

Multiplying  by  (x  —  1)  (x  —  2)  (x  —  3)  we  have 

1.  3x2 -12x  + 11=0. 

2.  This  is  not  so  easily  factored  as  in  the  first  problem;   hence, 
applying  the  formula  (§  271),  we  have 

X  =  -  ^^  ±  —  V(-  12)2  _  4  .  3  •  11 
2-3       2-3     ^         ' 


=  2±iV3. 
Check.     ^  H ^  + 


l±iV3      ±iV3       _l±^V3 

l-i_  1-i 

=  f  TiV3±  V3-f  TiV3  =  0. 

3.  Solve  the  equation  x^  -\-  2  x  =  0. 
This  factors  into  x  (x  +  2)  =  0,  whence  x  =  0  or  -  2. 
And,  in  general,  if  x  is  a  factor  of  every  term  of  an  equation,  x  =  0 
is  one  root. 


QUADRATIC   EQUATIONS. 


259 


EXERCISES.    CXX. 

Solve  the  following : 

1. 

1                  2                  13 

x  +  l      1-x      4.x -1 

2. 

3              2               1 

3-x      2-x      l-3ic 

3. 

2cc4-l      x-j-1      x-6 
x  +  1        x  +  2~  x-1 

4. 

4:X          X  -{-  1      X  -\-  5 
2ic-l          X          £c  +  4 

5. 

111 

a  —  X       a  —  2x       a  —  5x 

6. 

^              ^      +      1          2  =  0. 
x^-1       a;-lx  +  l 

7. 

2    2      ^      16      34   2       76          5 
3^       2      15~69^       115^  "^6 

8. 

x-2a      x-3b       x''-6ab_ 
2a              3b                6ab 

9.    V2-a;  +  V3  +  a?  -  Vll  +  ic  =  0. 

10.    (1  +  2  x)^  -  (3  +  x)*  +  (2  -  a:)^  =  0. 

6-\-5x   __  3x-4       5-7  X  _  :§!  ^  ^ 
^^'    4(5-x)       5(5+x)  ~^25-a;2      105 


4  (2  —  -\/x)       Vx  —  35   , 
12.    -^= ^  = -  + 


3x' 


13. 


Va;  +  a3         2  +  Vx      4(  V^  +  ic)  (2  +  Vx) 
4(2  +  V^)  _  ■\/x-[-x  3x^ 


+ 


V^  -  cc         2  -  Vic      4  (  Vic  -  ir)  (2  -  V^) 


260  ELEMENTS  OF  ALGEBRA. 

II.     DISCUSSION  OF  THE  ROOTS. 

273.  The  number  of  roots.  The  roots  of  the  equation 
ax^  +  6ic  +  c  =  0  have  been  shown  to  be 

2a      2a 

This  shows  that  every  quadratic  equation  has  two  roots. 

It  is  also  true  that  no  quadratic  equation  has  more  than 
two  different  roots. 

For,  suppose  the  equation  x^  -\-  px  -\-  q  =  0  has  three  dif- 
ferent roots,  Ti,  rg,  rg.  Then  by  substituting  these  for  x 
we  have 

1.  ri^  +  pi\  -{-  q  =  0, 

2.  r^^+pr^  +  q  =  (), 

3.  r-g^  +  pr^  +  2-  =  0,  whence 

4.  ri^  —  rg^  +  i?  (ri  —  r^)  =  0. 
Dividing  by  r^  —  r^,  which  by  hypothesis  9^  0, 

5.  ri  +  ^2  +  jp  =  0. 
Similarly,  taking  equations  2  and  3, 

6.  r2  4-^3+i?  =  0, 

7.  .-.  ri  —  T-g  =  0,  by  subtracting.    Ax.  3 

But  this  is  impossible  because,  by  hypothesis,  7\  ^  r^. 
Hence,  it  is  impossible  that  the  equation  shall  have  three 
different  roots,  and  so  for  any  greater  number. 

It  must  be  observed,  however,  that  a  quadratic  equation 
need  not  have  two  different  roots.     For  example,  the  equa- 

*^°^  x^-4.x  +  4.  =  0 

reduces  to  (x  —  2)(x~2)=  0, 

and  the  roots  are  2  and  2;  that  is,  the  equation  has  two 
roots,  but  they  are  equal. 


QUADRATIC   EQUATIONS. 


261 

4ac  is 


274.    The  nature  of  the  roots.     The  expression  V^ 
jailed  the  discriminant  of  the  quadratic  equation 

ax^  +  ^a;  +  c  =  0. 
In  this  discussion  a,  h,  c  are  supposed  to  be  real. 
If  the  discriminant  is  positive,  the  two  roots  are  real  and 
mequal. 

For  then  —  77—  ±  - —  V^^  —  4  ac  can  involve  no  imaginary. 

2a      2a  ^        J 

In  particular,  if  the  discriminant  is  a  perfect  square,  the 
two  roots  are  rational. 


For  then  V^^  —  4  ac  is  rational. 

If  the  discriminant  is  zero,  the  two  roots  are  equal. 


For  then 


Y-a^2-a^' 


Aac  = 


2a 


±0. 


In  this  case,  —  7--  is  called  a  double  root. 

2a 


If  the  discriminant  is  negative,  the  two  roots  are  complex. 
For  then 


2a     2a 


4c  ac  contains  the  imaginary 
I  V^>2  _  4  ac. 

Since  the  two  complex  roots  enter  together  the  instant 
that  y^  becomes  less  than  4  ac,  we  see  that  complex  roots 
iter  in  pairs. 
For  example,  in  the  equation 

the  roots  are  real,  since  3^  —  4  (—  7)  is  positive. 

In  2a;2  +  a;-3  =  0 

the  roots  are  rational,  since  1  —  (—  24)  is  a  perfect  square. 

In  3ic2^2a;  +  l  =  0 

bhe  roots  are  complex,  since  4  —  12  is  negative. 


262  ELEMENTS   OF  ALGEBRA. 

275.    Since    the   equation   ax^  -^  hx  -\-  c  =.  ^  has   for  its 

roots  —  TT-  +  ^r-  V^>^  —  4  ac  and  — p—  V^*^  —  4  ac,  it 

2a      2a  2  a      2a  ' 

follows  that 

Hence,  any  quadratic  function  of  x  can  he  factored 

1.  In  the  domain  of  rationality^ 

if  the  discriminant  is  square  ; 

2.  In  the  domain  of  reality, 

if  the  discriminant  is  positive  ; 

3.  In  the  domain  of  comjdex  numbers, 

if  the  discriminant  is  negative  ; 

4.  Into  two  equal  factors, 

if  the  discriminant  is  zero. 

Illustrative  problems.     1.  What  is  the  nature  of  the  roots 
of  the  equation  ic^  +  cc  +  l  =  0? 

•.•  6^  _  4  c^c  =  1  —  4  =  —  3,  the  two  roots  are  complex. 

2.  What  is  the  nature  of  the  roots  of  the  equation 

•.•  62  -  4  ac  =  36  -  36  =  0,  the  two  roots  are  equal. 

3.  What  is  the  nature  of  the  roots  of  the  equation 

4ic2  +  8aj  +  3  =  0? 
•••  62  —  4  ac  =  64  —  48  =  16,  the  roots  are  real,  unequal,  and  rational. 

4.  Can  fix)  =  hx^  ^Zx -1  \y^  factored  ? 

•••  ft'^  —  4  ac  =  9  +  140  =  149,  which  is  not  a  square,  /(x)  cannot  be 
factored  in  the  domain  of  rationality. 


QUADRATIC   EQUATIONS. 


263 


EXERCISES.    CXXI. 

What  is  the  nature  of  the  roots  of  equations  1-10  ? 


1.  5  a;2  +  1  =  0. 

3.  x""  -  X  +  1  =  0. 

5.  Sx^-\-x  +  7  =  0. 

7.  7a;2-a;-3  =  0. 

9.  i  x"^  -\- X  +  1  =  0. 


2.  a^x^  -\-  I  —  ax  =  0. 

4.  2  a;2  -  a;  -  20  =  0. 

6.  3x''-h4:X  +  5  =  0. 

8.  a^2_|_50a:4.625  =  0. 

10.  12a^2_i2a:  +  3  =  0. 


Of  the  following  functions  of  x  select  those  which  can  be 
itored  in  the  domain  of  rationality  and  factor  them. 


11.  3a;2_7. 

13.  6x^-^x  —  l. 

15.  7x''  +  2x-6. 

17.  6  ^2  _^  7  a: -3. 

19.  2x^-^3x-4:, 

21.  40x2  +  34^  +  6. 

23.  80x2  +  70^  +  60. 

25.  65x2-263x-42. 


12.  2x2  +  7x  +  3. 

14.  2ic2-5x  +  3. 

16.  55x^-27x-\-2. 

18.  llx2-23ic  +  2. 

20.  132  tt^  +  51 «  _  21. 

22.  121.r2  +  llx  +  12. 

24.  56x2  +  113^  +  56. 

26.  105x2 -246x  + 33. 


Reduce  the  following  to  the  form  ax^  +  &x  +  c  =  0,  and 
}tate  the  nature  of  the  roots : 


27. 


x  + 


Vx 


X 


-  V 


4. 


X 


-^-1 


0. 


28     (^  +  «)^       ^  +  ^  =  3 
(«  —  ^•)2       a  —  b 

2x  +  b      4:X  —  a 
30. .  =  0. 


31.    2 


1       (x  -  1)^ 


32. 


a 

Vx 


2x-b 
20-  V^ 


Vx  — 5  Vx 


=  3. 


264  ELEMENTS   OF  ALGEBRA. 

276.   Relation  between  roots  and  coefficients.     The  roots  of 
the  equation  x^  +2)x  +  q  =  0  are 


P 


iK2  =  -  ^  -  i  Vp2  -4,q, 

Their  sum  is  x^  +  x^  =  —  p, 

and  their  product    XiX^  =  (  ~  ^  )  ~  (i  V^^  —  4  g')^ 

_  ^^      pp-  —  ^q 
~T  4 

That  is,  m  an  equation  of  the  type  x^  +  px  +  q  =  0, 

1.  The  sum  of  the  roots  is  the  coefficient  of  x  with  the 
sign  changed; 

2.  The  product  of  the  roots  is  the  absolute  term. 

These  relations  evidently  give  a  valuable  check  upon  our 
solutions.  Any  solution  which  contradicts  these  laws  is 
incorrect. 

E.g.,  if  the  student  finds  the  roots  of  the  equation  x^  —  x  —  30  =  0 
to  be  —  6  and  5,  there  is  an  error  somewhere  in  the  solution,  because 
their  sum  is  not  the  coefficient  of  x  with  its  sign  changed. 

EXERCISES.     OXXII. 

Solve  the  following,  checking  by  the  above  laws. 
1.    a;2  +  1  zz.  0.  2.    ic2  -  1  =  0. 

3.    £C2  +  iC  =  0.  4.    x2  -  ic  -  1  =  0. 

5.    cc2-6a;  +  8  =  0.  Q.    x^-x-2  =  0. 

1.   x''-6x  +  4.  =  0.  8.    ic2  -  17  a;  +  16  =  0. 

9.    x^-12x  +  21  =  0.  10.    a;2  + 24a; +  144  =  0. 


QUADRATIC   EQUATIONS. 


265 


277.    Formation  of  equations  with  given  roots.     Since  if 

X  =  Ti  and         X  =  r^, 
len  X  —  ri  =  0,     "  x  —  r2  =  0, 

[and  hence  (x  —  r-^(x  —  T^  =  0,  a  quadratic  equation ;  there- 
of ore  it  is  easy  to  form  a  quadratic  equation  with  any  giyen 
roots. 

E.g.^  to  form  the  quadratic  equation  whose  roots  are  2  and  —  3. 

1.  •.•  X  =  2,       .-.  X  -  2  =  0. 

2.  •••  X  =  -  3,  .-.  X  +  3  =  0. 

3.  .-.  (x  -  2)  (x  +  3)  =  0,  or  x2  +  X  -  6  =  0. 
Similarly,  to  form  the  equation  whose  roots  are  i  ±  i  i. 

1.  •.•  x  =  i  +  ii,  .-.  x-i-ii  =  0. 

2.  V  x=:i-ii,  .-.  X- J  +  ii  =  0. 

3.  .-.  (x  —  i  -  i  i)  (x  —  i  +  i  i)  =  0,  and  this  may,  if  desired,  be 
dtten  in  the  form 

X2  _  X  +  t\  =  0, 

16x2-16x4-5  =  0. 


EXERCISES.    CXXIII. 

Form  the  equations  whose  roots  are  given  below. 

2.  V2,  Va 

4.    V2,  -  3. 

6.    -  7,  -  8. 

8.    ^-Vh,\Va. 
a  b 

10.    3  +  2*,  3  -2i. 

12.    5  +  3  i,  5  —  3  i. 

1.      14.    a  +  2  V^,  a~2- 


1. 

§>!• 

3. 

i,  —  i. 

5. 

3,  -  11. 

7. 

a         a 

-2'-2' 

9. 

-a  ±2  hi. 

11. 

-i±i'iV3. 

13. 

a    / — -        a 

-1. 


266  ELEMENTS   OF   ALGEBRA. 

IIL     EQUATIONS   REDUCIBLE  TO   QUADRATICS. 

278.  Thus  far  the  student  has  leai-ned  how  to  solve 
any  equation  of  the  first  or  second  degree  involving  one 
unknown  quantity,  and  simultaneous  equations  of  the  first 
degree  involving  several  unknown  quantities. 

It  is  not  within  the  limits  of  this  work  to  consider  gen- 
eral equations  of  degree  higher  than  the  second.  It  often 
happens,  however,  that  special  equations  of  higher  degree 
can  be  solved  by  factoring,  as  already  explained,  or  by 
reducing  to  quadratic  form. 

A  few  of  the  more  common  cases  will  now  be  considered, 
some  having  already  been  suggested  in  the  exercises. 

279.  The  type  ax^"  +  bx"  +  c  =  0.  This  is  a  quadratic 
in  aj",  and  (§  269) 


whence 


X"  =  —  7—  ±  --—  V^/"-^  —  4  ac, 
Za      la 


^      Za      la 


Illustrative  problems.      1.  Solve  the  equation 
a;6  +  10a.-3  +  16--0. 

This  is  a  quadratic  in  x^  and  is  easily  solved  by  factoring. 
■    ...  (x3  +  8)  (x3  +  2)  =  0, 
.-.  x3  =  -  8,  or   -  2. 
.-.x   =  -  2,  or  -  V2. 

CJieck  for  x  =  -  \/2.     4  -  20  +  16  =  0. 

We  might  also  solve  by  the  above  formula,  thus : 

3/ ■ 

x=  V-  5  ±  i  VlOO  -  64 
=  V3^,  or  V^s  =  -2. 


QUADRATIC   EQUATIONS. 


267 


2.  Solve  the  equation  x^  -\-  x  -^1  -\ 1 — ^  =  0. 

X       x^ 
This  may  be  arranged 

(x  +  -j   +(»  +  -]  —  1=0,  a  quadratic  in  x  +  - 


Solving  (§  270), 


X  + 


^  =  -i±iV5. 


x'^-{-i±i^b)x  +  l=0, 


id  (§  270) 


X  =  i{V5  -1  ±i  VlO  +  2  VS), 


i(-  V5-I  ±i  VlO-2  V5). 

3.  Solve  the  equation  x~^  -\-  x~^  —  2  =  0. 

Tliis  is  a  quadratic  in  x~^.     Solving  by  factoring, 
x~'^=  1,  or  -  2. 
1 


1,  or 


(-2)4 


Check  for  x 


(-2)2 +  (-2) -2=0. 


(-2)4 

If  (—  2)4  had  been  written  16,  there  would  appear  to  be  an  extra- 
leous  root,  but  by  writhig  it  (—  2)4  we  know  that  the  4th  root  is  —  2. 


4.  Solve  the  equation  cc'^  =  21  +  Va;^  —  9. 

This  may  be  arranged 

(x-2  -  9)  -  (x2  -  9)^  -  12  =  0. 
The  solution  often  seems  easier  if  y  is  put  for  the  unknown  expres- 
ion  in  the  quadratic.     Here,  let  y  =  (x^  —  9)*.     Then 
y-^-y -12  =  0, 
(2/-4)(2/  +  3)  =  0, 
rhence  ?/  =  4,  or  —  3. 

x2-9=  10,  or  (-3)2, 
id  x2  ^  25,  or  9  +  (-3)2, 

id  X  =  ±  5,  or  ±  V9  +  ( -  3)2. 


Check  for  X  =  ±  V9  +  (-3)2.     9  +  (-  3)2  =  21  +  V9  +  (-  3)2  -  9, 
)r  18  =  21  -  3,  because  V(-  3)2  =  -  3.     If  the  (-3)2  were  written  9, 
fthere  would  appear  to  be  an  extraneous  root. 


268  ELEMENTS  OF  ALGEBRA. 

6.  Solve  the  equation  (x^  +  x  -{-  3)  (x^  -^  x  -\-  5)  =  35. 

In  equations  of  this  kind  there  is  often  an  advantage  in  letting  y 
equal  some  function  of  x.     Here,  let  2/  =  x^  -f  x  +  3.     Then 

y(2/  +  2)  =  36, 
or  2/2  +  2  2/  -  35  =  0, 

or  {y  +  7)  (y  -  5)  =  0, 

whence  2/  =  —  7,  or  5. 

Hence,  x^  +  x  +  3  =  —  7,  or  5, 

and  each  of  these  equations  can  be  solved  for  x. 

It  would  answer  just  as  well  tolet  y  =  x^  +  x  -\-  5,  in  which  case  we 
should  have  {y  —  2)y  =  35. 

EXERCISES.    CXXIV. 

Solve  the  following : 


1.    Vic  -  1  =  cc  —  1.  2.  a;-ic*-20  =  0. 

3.    7x-4.x^  -20  =  0.  4.  x^  +  x^-20  =  0. 

5.    a;«-28a;«  +  27  =  0.  6.  7  x^ -\- x^  -  350  =  0. 

7.  x^  +  x  —  6x^  =  0. 

8.  x^  +  5x-l  =  - ^ -• 

x^  +  5x  +  l 

9.  (ic2  +  3)2  +  (x^  -I-  3)  _  42  =  0. 

10.  x-(a  +  b)x^-2a(a-b)  =  0. 

11.  (x^-\-2x-\-3)(x^  +  2x-h6)  =  -2. 

12.  (x2  +  3a;-4)(ic2  +  3aj  +  2)  +  8  =  0. 

13.  V^/(21- V^)  +  (21- V^)/V^  =  2.5. 
^4  1  ■  1 6  ^ 


QUADRATIC   EQUATIONS. 


269 


280.    Radical  equations  have  already  been  discussed  (§  191) 
the  special  case  in  which  they  lead  to  simple  equations, 
id  several  problems  have  been  given  in  connection  with 
^the  study  of  quadratics. 

Whenever  they  lead  to  quadratic  equations  their  solution 
IS  possible,  and  a  few  cases  somewhat  more  elaborate  than 
[those  already  given  will  now  be  considered. 


Illustrative  problems.     1.   Solve  the  equation 
2a;2_|.3^_3V2ic2_^3:c-4-2  = 
1.   This  may  be  arranged 


0. 


2x2  +  3x-4-3V2x2  +  3x-4  +  2  =  0. 

2.  Let  y  =  V2x2  +  3x-4. 

3.  Then  y^  -Sy  +  2  =  0. 

4.  .-.  (y-2){y-l)  =  0. 

5.  .-.  y  =  2,  or  1. 

6.  .-.  2x2  +  3x-4  =  2,  or  1, 
two  quadratic  equations  in  x,  which  give 

X  =  -  I  ±  i  V57,  1,  or  -  |. 
Check  for  X  =  -  |.     2.5  _  j^  _  3  _  2  =  0. 

2.  Solve  the  equation  x  —  1  =  2  -{-  2x~^. 

1.  This  may  be  written 

(V^-i)(V^  +  i)-'^^  +  ^^^o. 
Vx 

2.  Or  (V^+ l)('Vx- 1 -4=')  =  0. 
^  Vx/ 

3.  .-.  Vx+1  =  0,  and  Vx  =  -1,  andx 
land  X 


2 

Vx 


0, 


(-1)2,  orVx 
Vx  —  2  =  0,  a  quadratic  in  Vx. 

4.  .-.  (Vx-2)(V^+1)  =  0. 

5.  .-.  Vx  =  2,  and  x  =  4,  or  Vx  =  —  1,  and  x  =  {—  1)2. 

.-.  there  are  three  roots,  two  being  alike,  4,  (—  1)2,  (—  1)2.  All 
three  are  easily  seen  to  check.  The  reason  for  writing  ( —  1)2  instead 
of  +  1  is  explained  on  p.  267,  exs.  3  and  4. 


270  ELEMENTS  OF  ALGEBRA. 


3.  Solve  the  equation  Va;  +  3  —  Vic  +  8  =  5  V^. 


1.  2a;+ 11 -2  V(x  +  3)(x  +  8)  =25x.  Ax.  8 

2.  .-.  -  2  Vx2  +  11  X  +  24  =  23x  -  11.  Ax.  3 

3.  .-.  21  x'^  —  22  X  +  1  =  0,  squaring,  etc. 

4.  .-.  (21  X  -  1)  (x  -  1)  =  0,  and  x  =  ^\,  or  I. 

In  checking,  each  root  is  found  to  be  extraneous.  This  might  have 
been  anticipated  because  in  squaring  the  first  member  of  step  2  the 
( —  2)2  was  called  4,  and  hence,  when  the  result  was  placed  under  the 
radical  sign  for  checking,  and  the  root  taken  as  positive,  a  failure  to 
check  was  natural. 

Had  the  original  equation  been  Vx  +  3  +  Vx  +  8  =  5  Vx,  tlie  root 
1  would  have  checked  ;  had  it  been  —  Vx  +  3  +  Vx  +  8  =  5  Vx,  the 
root  2X  would  have  checked. 


EXERCISES.    CXXV. 

Solve  the  following : 

1. 

Va;  +  3  -  V;r  -  4  -  1  =  0. 

2. 

x^J^x  =  4.-i~  VlO  -  X''  -  X. 

3. 

Vl  +4^;- Vl  -4x  =  4  Vaj. 

4. 

Vx2  -8x-{-31+(x-4.y  =  5. 

5. 

x'^-\-5x-W  =  Wx^-\-5x  +  2. 

6. 

7. 

Va;  -  2  +  V3  +  ic  -  VlO  +  ^  =  0. 
xi(x^  _  l)i  _  2  x  (x^  -  1)^  -  ^  ==  0. 

8. 

Vl  +  2  a^  -  V4  +  a;  +  V3  -  a-,  =  0. 

9. 

Vcc  +  8  +  Vx  -  6  -  V8  a^  -  10  =  0. 

10. 

V4a;-2  +  2V2-ic-  Vl4  -4.x  =  0. 

11. 

3  Vic^  -  7  X  +  12  =  V7  •  Vx-  -  7  ic  +  12. 

12.    V(ic  -  1)  (x  -  2)  +  V(a'  -  3)  (.X  -  4)  =  V2. 


QUADRATIC   EQUATIONS. 


271 


281.    Reciprocal  and  binomial  equations.     A  reciprocal  equa- 

jtion  is  an  equation  in  which  the  coef&cients  of  the  terms 

jquidistant  from  those  of  highest  and  lowest  degree,  respec- 

Lvely,  have  the  same  absolute  value  and  have  the  same 

jigns  throughout  or  opposite  signs  throughout. 

E.g.,  the  following : 

ic2  -  1  =  0, 

ax^  +  hx^  —  bx  —  a  =  0, 

ax^  —  bx^  -\-  cx^  —  bx  -^  a  =  0, 

x^  -{-  x^  -h  x^  -i-  x'^  +  X  +  1  =  0. 

They  are  called  reciprocal,  because  they  are  unaltered 
rhen  for  the  unknown  quantity  is  written  its  reciprocal. 

Kg.,  when  -  is  written  for  x  in  the  equation 


becomes 


ax^  -\-  bx  -\-  a  =  0, 

X"^         X 


rhich,  by  multiplying  both  members  by  x^,  reduces  to 

a  -{-  bx  -\-  ax^  =  0, 
le  original  equation. 

282.    Since  x  can  be  replaced  by  -  >  the  roots  of  reciprocal 

mations  enter  in  pairs,  each  root  being  the  reciprocal  of 
the  other  root  of  that  pair,  excepting  the  two  roots  + 1  and 
1,  each  of  which  is  its  own  reciprocal. 

E.g.,  x"^  -\- X  -\- 1  =  0  has  for  its  roots 

^1  =  -  i  +  i  *  Va, 

x^  =  -^-^i  Vs, 

id  each  is  the  reciprocal  of  the  other,  because  their  prod- 
ict  is  1  (§  162). 


272  ELEMENTS  OF  ALGEBRA. 

So  ic^  +  1  =  0  has  for  its  roots  the  reciprocals  i  and  —  i. 
Similarly  in  the  case  of  x^  —  2  x^  —  2  x  -\- 1  =  0.     Here 

x^  +  l-2x(x-\-l)  =  0, 

whence  (x  -{- 1)  (x^  —  x  -}- 1  —  2  x)  =  0, 

and  therefore  a?  +  1  =  0,  and  x  =  —  1, 

or  cc2  -  3  ic  +  1  ==  0,  and  a;  =  I  ±  1  Vs. 

In  this  case,  f  +  ^  V5  and  |  —  ^  Vs  are  reciprocals, 
because  their  product  is  1  (§  162),  and  the  other  root, 
—  1,  is  its  own  reciprocal.  And  in  general,  in  the  case  of 
reciprocal  equations  of  odd  degree,  one  root  is  always  its 
own  reciprocal. 

This  is  seen  in  the  case  of  x^  —  1  =  0. 

283.  Reciprocal  equations  can  often  be  reduced  to  equa- 
tions of  lower  degree  by  the  factoring  method  set  forth  in 
the  preceding  example,  or  by  dividing  by  some  power  of  the 
unknown  quantity,  as  in  the  following  case : 

Solve  x^  -j-  x^  +  x^  -{-  X  +  1  =  0. 
Divide  by  x^,  and  -^       2 

X2  +  X  +  1+-+  —  =  0, 

X       x2 
an  equation  already  considered  (§  279). 

It  reduces  to  (  x  +  -  j  +  (  x  +  -  j  —  1  =  0, 

a  quadratic  in  x  -j Solving  for  x  +  - ,  we  have 

X  X 

x  +  ^=-i±iV5.  §270 

two  quadratics  in  x. 

These  equations  may  now  be  solved  for  x,  each  giving  two  values. 
The  final  roots  are  four  in  number,  as  would  be  expected.  They  are 
given  on  p.  267,  and  in  more  complete  form  on  p.  273. 


QUADRATIC   EQUATIONS. 


273 


284.  Equations  of  the  form  x""  -\- p  =  0  are  called  bino- 
[mial  equations.  In  this  case,  no  restriction  is  placed  on  p ; 
jit  may  be  positive,  negative,  integral,  fractional,  real,  imag- 
finary,  etc. 

The  solution  of  binomial  equations  in  which p  =  ±l  evi- 
dently depends  upon  the  solution  of  a  reciprocal  equation. 

E.g.,  x5-l=0 

luces  to  (x  -  1)  (x4  +  x3  +  x2  +  X  +  1)  =  0, 

whence  x  —  1  =  0,  and  x  =  1, 

3r  x*  +  x3  +  x2  +  X  +  1  =  0, 

a  reciprocal  equation,  the  one  just  considered  in  §  283,  with  four  roots. 

Since  if  cc^  —  1  =  0,  or  ic^  =  1,  a:  is  the  fifth  root  of  1,  and 
3mce  cc^  -  1  =  0  has  5  roots  (§§  279,  283),  viz.: 

Xi  =  l, 

CC2  =:  1  ( V5  -  1  +  ^  VlO  +  2  V5), 
xs  =  i(V5  -  1  -  ^VlO  +  2  VS), 


x,  =  l(--^5-l+  ^  VlO  -  2  VS), 
x,  =  l(~^5-l-i  VlO  -  2  V5), 

therefore,  there  are  5  fifth  roots  of  1. 

Similarly,  there  are  2  square  roots  of  any  number,  3  cube 
roots,  •  •  ■  n  nth.  roots. 

Thus  the  two  square  roots  of  1  are  evidently  +1,  —  1, 
jwhich  may  be  obtained  by  extracting  the  square  root 
[directly  or  by  solving  the  equation  o;^  —  1  ==  0. 

The  three  cube  roots  are  readily  found  by  solving  the 
[equation  cc^  —  1  =  0. 

Here  a:;^  —  1  =  0 

[leads  to     (x  —  1)  (x"^ -\- x  -{- 1)  =  0, 
whence  a;  —  1  =  0,  and  x  =  1,' 

)X  x^-\-x  +  l  =  0,  solved  in  §  282. 


274  ELEMENTS   OF   ALGEBRA. 

EXERCISES.     CXXVI. 

Solve  the  following : 

1.  2i»2  +  5cc  +  2  =  0. 

2.  £C»  +  cc^  +  ic  4- 1  =  0. 

3.  10a;2-29cc  +  10=:0. 

4.  2x^-3x^-Sx-\-2  =  0. 

5.  x^  +  x^  —  4.x^  -}-  X  +  1  =  0. 

6.  x^  -  x^  —  4.x^  —  X  -\-  1  =  0. 

7.  x^  +  4:X^  +  2x^  +  4.x  -\-l  =  0. 

8.  ic*  -  5  ic^  +  f  I  a:^  -  5  cc  +  1  =  0. 

9.  X*  —  I  ic^  —  -y-  cc^  —  I  X  +  1  =  0. 

10.  2x^-9x^  +  Ux''-9x  +  2  =  0. 

11.  12  a;* +  4x3 -41x^  +  40; +  12  =  0. 

12.    x^  +  1  =  0.  13.    x^  +  1  =  0.  14.    x^  -  1  =  0. 

15.  What  are  the  2  square  roots  of  1  ?  the  3  cube  roots  ? 
the  4  fourth  roots  ? 

16.  What  are  the  3  cube  roots  of  8  ? 

17.  What  are  the  6  sixth  roots  of  1  ? 

18.  Show  that  the  product  of  any  two  of  the  fourth  roots 
of  1  equals  one  of  the  four  roots^  and  that  the  cube  of  either 
imaginary  root  equals  the  other. 

19.  Show  that  the  product  of  any  two  of  the  cube  roots 
of  1  equals  one  of  the  three  roots,  and  that  the  square  of 
either  complex  cube  root  equals  the  other. 

20.  Show  that  the  sum  of  the  2  square  roots  of  1,  the 
sum  of  the  3  cube  roots,  the  sum  of  the  4  fourth  roots,  the 
sum  of  the  5  fifth  roots,  are  all  equal  to  zero. 


QUADRATIC    EQUATIONS. 


275 


285.    Exponential  equations  have  already  been  considered 
§  205.      Only  in  certain  cases  can  tliey  be  solved  by 
lear  or  quadratic  methods. 

E.g.,  2-^:  8- =  16:1. 

This  may  be  written 

2x^  —  ^x  __  24 

rhence  x^  —  ^  x  =^  4c, 

jiving  a:  =  4,  or  —  1.     Each  result  checks. 

The  equation  2^  +  ^  +  4^  =  8  may  be  written 
2 .  2^  +  22^  =  8, 
)r  (2^)2  +  2  (2^)  -  8  =  0,  a  quadratic  in  2^. 

Hence,  solving,  2^  =  2  or,  —  4. 

If  2^  =  2,  X  =  1,  a  result  which  checks. 

If  2^  =  —  4,  we  cannot  find  x. 


EXERCISES.     CXXVII. 

Solve  the  following : 
1.    64^:  2- =  4. 
3.    2-.2-^'  +  i  =  2. 
5.    2^.16^  =  ^1^. 
7.    2  •  4^^^  =  2^^-^ 


9.    (4V8)"  =  23--  +  ». 


11. '^  =  1, 


13. 


m^^  =  1. 


15.    2-5 


9  .  r-fi-x .  9K 


2.  3"':8P  =  (3ii)'. 

4.  32^ -9^  =  27^' -3. 

6.  a'^:{a'=f  =  {ay. 

8.  (3-f-3-  =  27i^ 

10.  2-6^*  +  '^  =  3^^*- 2^+ 

12.  9-^-9-^  =  ^. 

14,  a^'^^' .  (ay-' =  \' 

2/25^-^ 


276  ELEMENTS   OF  ALGEBRA. 

IV.     PROBLEMS    INVOLVING   QUADRATICS. 

Illustrative  problems.     1.    What  number  is  0.45  less  than 
its  reciprocal  ? 


L    Let 

X  =  the  number. 

2.   Then 

x  =  --  0.45. 

X 

3.   .-. 

aj2  +  o.45x- 

-1  =  0. 

4.   .-. 

x=  -  0.225  X  ±  0.5  Vo.2025  +  4 

=  0.8,  or  -1.25. 

Check.     0.8 

=  L25-0.45. 

-1.25=  -0.8-0.45. 

Hence,  either  result  satisfies  the  condition.  But  if  the 
problem  should  impose  the  restriction  "in  the  domain  of 
positive  numbers,"  —  1.25  would  be  excluded ;  if  "  in  the 
domain  of  negative  numbers,"  0.8  would  be  excluded ;  if 
"  in  the  domain  of  integers,"  both  results  would  be  excluded 
and  no  solution  would  be  possible. 

2.  A  reservoir  is  supplied  with  water  by  two  pipes.  A,  B. 
If  both  pipes  are  open,  |i  of  the  reservoir  will  be  filled  in 
2  mins. ;  the  pipe  A  alone  can  fill  it  in  5  mins.  less  time 
than  B  requires.  Find  the  number  of  minutes  in  which 
the  reservoir  can  be  filled  by  A  alone. 

1.  Let  X  =  the  number  of  minutes  required  by  A. 

2.  Then  ■  a;  +  5  =  "  "  "  B. 

3.  Then  -  =  part  filled  by  A  in  1  miu. , 

and  =         "  "   B  "  1  min. 

x  +  5 

4..-.  ?+     ' 


X      ic  +  5 

5.  .-.       11  x2  +  7  X  -  120  =  0,  or  (x  -  3)  (11  x  +  40)  =  0. 

6.  .-.  X  =  3,  or  -  f  ft. 


QUADRA. :C   EQUATIONS. 


277 


Here  each  root  satisfies  the  equation  ;  but  the  conditions 
)i  the  problem  are  such  as  to  limit  the  result  to  the  domain 
positive  real  numbers.     Hence,  —  f  ^,  being  meaningless 
this  connection,  is  rejected. 

3.  The  number  of  students  in  this  class  is  such  as  to 
Satisfy  the  equation  2  a;^  —  33  a?  =  140.  How  many  are 
there  ? 


1. 


2  x2  -  33  a;  -  140  =  0. 

(X  -  20)  (2  X  +  7)  =  0. 

X  =  20,  or  -  |. 


Here,  too,  the  conditions  of  the  problem  are  such  as  to 
it  the  result;  this  time  to  the  domain  of  positive  inte- 
gers.    Hence,  "  —  |  of  a  student,"  being  meaningless,  is 
rejected. 


4.  A  line,  AB,  3  in.  long,  is  produced  to  P  so  that  the 
rectangle  constructed  with  the  base  AF  and  the  altitude 
»P  has  an  area  14.56  sq.  in. 
rind  the  length  of  BF. 


P' 


1.  Let  X 

=  the  number  of  inches  in  BP. 

2.  Then  the  area  R 

=  (3  +  x)x  =  14.56. 

3.  .•.x2  +  3x- 14.56  =  0. 

4.  .-.  X  =  2.6,  or  -  6.6. 


Here  we  are  evidently  not 

lited  as  in  probs.  2  and  3. 
?he  negative   root   may  be 

iterpreted  to  mean  that  AB  is  produced  to  the  left. 
?P'  is  —  5.6  in.,  i.e.,  5.6  in.  to  the  left,  and  the  rectangle 
jcomes  E',  which  is,  however,  identically  equal  to  B. 


278  ELEMENTS   OF  ALGEBRA. 

EXERCISES.     CXXVIII. 

In  each  exercise  discuss  the  admissibility  of  both  roots. 

A.     Eelating  to  Numbers. 

1.  What  number  is  y\  of  its  reciprocal  ? 

2.  What  number  is  -^^  greater  than  its  reciprocal  ? 

3.  What  is  the  number  which  multiplied  by  f  of  itself 
equals  1215  ? 

4.  Separate  the  number  480  into  two  factors,  of  which 
the  first  is  |  of  the  second. 

5.  The  sum  of  a  certain  number  and  its  square  root  is 
42.     Required  the  number. 

6.  Find  a  number  of  which  the  fourth  and  the  seventh 
multiplied  together  give  for  a  product  112. 

7.  One-fourth  of  the  product  of  f  of  a  certain  number 
and  I  of  the  same  number  is  630.     Find  the  number. 

8.  The  square  of  5  more  than  a  certain  number  is 
511,250  more  than  10  times  the  number.  Required  the 
number. 

9.  The  product  of  the  numbers  2ic3  and  4cc6,  written 
in  the  decimal  system,  is  115,368.  What  figure  does  x 
represent  ? 

10.  Separate  the  number  3696  into  two  factors  such  that 
if  the  smaller  is  diminished  by  4  and  the  larger  increased 
by  7  their  product  will  be  the  same  as  before. 

11.  Of  three  certain  numbers,  the  second  is  f  of  the  first, 
and  the  third  is  |  of  the  second ;  the  simi  of  the  squares 
of  the  numbers  is  469.     What  are  the  numbers  ? 


QUADRATIC   EQUATIONS.  279 

B.     Relating  to  Mensuration. 
For  formulas  see  p.  172. 

12.  How   many    sides    has    a   polygon   which    has   54 
[iagonals  ? 

13.  The  area  of  a  rectangle  is  120  sq.  in.,  and  its  diagonal 
17  in.     Required  its  length  and  breadth. 

14.  The  base  of  a  triangle  of  area  16.45  sq.  in.  is  2.3  in. 
|iore  than  the  altitude.     Required  the  base. 

15.  The  length  of  a  rectangle  of  area  70  sq.  in.  is  3  in. 
)re  than  the  breadth.     Required  the  dimensions. 

16.  Divide  a  line  16  in.  long  into  two  parts  which  shall 
)rm  the  base  and  altitude  of  a  rectangle  of  63.96  sq.  in. 

17.  The  hypotenuse    of   a   right-angled    triangle   is  10 
and  one  of  the  sides  is  2  in.  longer  than  the  other. 

jquired  the  lengths  of  the  sides. 

18.  In  a  right-angled  triangle  one  of  the  sides  forming 
le  right  angle  is  6  in.,  and  the  hypotenuse  is  double  the 

ler  side.     Find  the  length  of  the  other  side. 

19.  A  square  and  a  rectangle  have  together  the  area  220 
sq.  in.  The  breadth  of  the  rectangle  is  9  in.,  and  the 
length  of  the  rectangle  equals  the  side  of  the  square. 
Required  the  area  of  the  square. 

k  20.  From  the  vertex  of  a  right  angle  two  bodies  move 
on  the  arms  of  the  angle,  one  at  the  rate  of  1.5  ft.,  and  the 
other  2  ft.,  per  second.  After  how  many  seconds  are  they 
50  ft.  apart  ? 


I 


21.    What  is  the  result  if,  in  the  preceding  example,  1.5, 
,  and  50  are  replaced  by  m.,  n,  d'i 


280  ELEMENTS   OF  ALGEBKA. 

22.  A  square  is  78  sq.  in.  greater  than  a  rectangle.  The 
breadth  of  the  rectangle  is  7  in.,  and  the  length  is  equal  to 
the  side  of  the  square.     Required  the  side  of  the  square. 

23.  If  the  sides  of  a  certain  equilateral  triangle  are 
shortened  by  8  in.,  7  in.,  and  6  in.,  respectively,  a  right- 
angled  triangle  is  formed.  Required  the  length  of  the  side 
of  the  equilateral  triangle. 

24.  If  two  sides  of  a  certain  equilateral  triangle  are 
shortened  by  22  in.  and  5  in.,  respectively,  and  the  third 
is  lengthened  by  3  in.,  a  right-angled  triangle  is  formed. 
Required  the  length  of  a  side  of  the  equilateral  triangle. 

25.  On  an  indefinite  straight  line  given  two  points,  A 
and  B,  d  units  apart,  to  find  on  this  line  a  point,  F,  such 
that  AP^  =  BP '  AB.  Draw  the  figure  showing  the  posi- 
tions o-f  the  two  points.  (This  is  the  celebrated  geometric 
problem  of  "  The  Golden  Section.") 

26.  Four  places.  A,  B,  C,  D,  are  represented  by  the 
corners  of  a  quadrilateral  whose  perimeter  is  85  mi.  The 
distance  BC  is  24  mi.,  and  CD  is  14  mi.  The  distance 
from  ^  to  D  by  the  way  of  B  and  C  is  y\  as  great  as  the 
square  of  the  distance  from  A  direct  to  D.  How  far  is  it 
from  Ato  B?    also  from  ^  to  D  ? 

27.  About  the  point  of  intersection  of  the  diagonals  of  a 
square  as  a  center,  a  circle  is  described ;  the  circumference 
passes  through  the  mid-points  of  the  semi-diagonals;  the 
area  between  the  circumference  and  the  sides  of  the  square 
is  971.68  sq.  in.  Required  the  length  of  the  side  of  the 
square.     (Take  tt  =  3.1416.) 

28.  A  mirror  56  in.  high  by  60  in.  wide  has  a  frame  of 
uniform  width  and  such  that  its  area  equals  that  of  the 
mirror.     What  is  the  width  of  the  frame? 


QUADRATIC   EQUATIONS.  281 

C.     Eelating  to  Physics. 

29.  If  a  bullet  is  fired  upward  with  a  velocity  of  640  ft. 
[per  sec.,  the  number  of  seconds  elapsing  before  it  strikes 

the  earth  is  represented  by  t  in  the  equation  0  =  320  t  —  ^gf'^ 
^in  which  ^  =  32  ft.     Find  t. 

30.  Two  points,  A  and  i?,  start  at  the  same  time  from  a 
fixed  point  and  move  about  the  circumference  of  a  circle  in 
)pposite  directions,  each  at  a  uniform  rate,  and  meet  after 

sees.     The  point  A  passes  over  the  entire  circumference 

b  9  sees,  less  time  than  B.      Required  the  time  taken  by 

fA,  and  also  by  B,  in  passing  over  the  whole  circumference. 

31.  It  is  shown  in  physics  that  if  two  forces  are  pull- 
ing from  a  point,  P,  and  are  represented  in  direction  and 

tensity  by  the  lines  PA,  PB,  the  resultant  force  is  repre- 
ented  by  PC,  the  diagonal  of  their  parallelogram.  Two 
brces,  of  which  the  first  is  23  lbs.  greater  than  the  second, 

t  at  right  angles  from  a  point.     Their  resultant  is  37  lbs. 

equired  the  intensity  of  each  force. 

32.  Two  forces,  of  which  the  first  is  47  lbs.  less  than 
the  second,  act  at  right  angles  from  a  point.  Their  result- 
ant is  65  lbs.     Required  the  intensity  of  each  force. 

33.  It  is  proved  in  physics  that  if  a  body  starts  with  a 
velocity  ("initial  velocity")  of  u  ft.  per  sec,  and  if  this 
increases  a  ft.  per  sec.  (the  "  acceleration  "),  then  in  t  sees, 
the  space  s  described  i^  s  =  ut  -{-  ^  at^.  Suppose  the  initial 
velocity  is  40  ft.  per  sec,  and  the  body  moves  with  an  ac- 
celeration of  —  2  ft.  per  sec,  find  when  it  will  be  400  ft. 

from  the  starting  point. 

^^B'  34.  Suppose  a  body  starts  from  a  state  of  rest,  and  the 
^H  acceleration  is  18  ft.  per  sec,  find  the  time  required  to  pass 
^H  over  the  first  foot ;  the  second ;  the  third.     (See  ex.  33.) 

I 


282  ELEMENTS   OF   ALGEBRA. 

35.  Two  points,  A  and  B,  start  at  the  same  time  from  a 
fixed  point  and  move  about  the  circumference  of  a  circle 
in  the  same  direction,  each  at  a  uniform  rate,  and  are  next 
together  after  8  sees.  The  point  A  passes  over  the  entire 
circumference  in  18  sees,  less  time  than  B.  Required  the 
time  taken  by  A  iji  passing  over  the  whole  circumference. 

36.  It  is  shown  in  physics  that  if  h  =  the  number  of  feet 
to  which  a  body  rises  in  ^.secs.  when  projected  upward 
with  a  velocity  of  it  ft.  per  sec,  then  h  =  ut  —  \  gf-,  where 
g  =  32.  Find  the  time  that  elapses  before  a  body  which 
starts  with  a  velocity  of  64  ft.  per  sec.  is  at  a  height  of 
28  ft. 

37.  A  body  is  projected  vertically  upward  with  a  velocity 
of  80  ft.  per  sec.  When  will  it  be  at  a  height  of  64  ft.  ? 
(See  ex.  36.) 

D.     Miscellaneous. 

38.  A  reservoir  can  be  filled  by  two  pipes,  A  and  B,  in 
9  mins.  when  both  are  open,  and  the  pipe  A  alone  can  fill 
it  in  24  mins.  less  time  than  B  can.  Required  the  number 
of  minutes  that  it  will  take  A  alone  to  fill  it. 

39.  A  reservoir  has  a  supply  pipe,  A,  and  an  exhaust  pipe, 
B.  A  can  fill  the  reservoir  in  8  mins.  less  time  than  B  can 
empty  it.  If  both  pipes  are  open,  the  reservoir  is  filled  in 
6  mins.  Required  the  number  of  minutes  which  it  will 
take  to  fill  it  if  A  is  open  and  B  is  closed. 

40.  Two  travelers,  A  and  B,  set  out  at  the  same  time 
from  two  places,  P  and  Q,  respectively,  and  travel  so  as  to 
meet.  When  they  meet  it  is  found  that  A  has  traveled 
30  mi.  more  than  B,  and  that  A  will  reach  ()  in  4  das.,  and 
B  will  reach  P  in  9  das.  after  they  meet.  Find  the  dis- 
tance between  P  and  Q. 


QUADRATIC   EQUATIONS. 


283 


1.    Solve 


REVIEW   EXERCISES.     CXXIX. 
X^  +  111^  _  X 


2.    Solve  -  +  -  =  -  +  -■ 
2      a;      3      ic 


3.  Factor  (x""  +  x  -  ISy  -  49. 

4.  Factor  x^-6x^-37x  +  210. 

2x      ^   4tx  —  3       _       _ 

5.  Solve  7  H -^i 9  =  0. 


6.    Solve 


7.    Solve 


.r  —  4        X  -\- 1 

x-iy     fx- 
x  +  l)       \x  + 


2\2 

2 


a. 


X  —  h 


=  0. 


8.  Simplify 

^,  ,        /x2  -  11 

9.  Solve      — 

\    X'^  -{-  X 


X  —  b       X  —  a       cr  —  ax 

2x'-x-{-2      4  .r^  -  1 


4ic3  +  3x  +  2     2x-l 

^  +  19  Y_  3(2-^ 


11  ;       2  +  x 

10.  Solve  18  (x  +  iy{x  +  2)2  =  8  (x  -  3)2(ic  +  l)^. 

11.  Solve  {x  -3y-3{x-  2)^  +  3{x  -  If  -  x^  =  ^  - x. 

12.  Find  the  square  root  of 

13.  li  x"^  -\-xy+  z  =  0,  and  iv"^  -\-  luy  +  «  =  0,  where  x^w, 
prove  that  w  +  x  +  ?/  =  0.     (Subtract  and  factor.) 

14.  Find  the  lowest  common  multiple  of 
(^,2  ^  c''  -  a''  +  2hc)  {c  +  a  -  h) 


.aad 


{a^ 


^  ^-2hG)(a-\-h  +  c). 


CHAPTER   XV. 

SIMULTANEOUS    QUADRATIC    EQUATIONS. 

I.     TWO  EQUATIONS   WITH  TWO  UNKNOWN  QUANTITIES. 

286.  1.  When  one  equation  is  linear.  While  this  is  not 
a  case  in  simultaneous  quadratics,  since  one  equation  is 
linear,  it  forms  a  good  introduction  to  the  general  subject. 

In  this  case,  one  of  the  unknown  quantities  can  be  found 
in  terms  of  the  other  in  the  linear  equation,  and  the  value 
substituted  in  the  quadratic.  The  problem  then  becomes 
that  of  solving  a  quadratic  equation. 

E.g.^  to  solve  the  system    x  —  2  2/  =  3. 
0:2  -I-  2/2  =  26. 


(Why  ?) 


Here^ 

tve  have 

1. 

X  ■■ 

=  3  +  2  2/,  from  the  first  equation. 

2.    .-. 

(3  +  2^)2  +  2/2  = 

:26. 

3.    .-. 

52/2 +  12  2/ -17  = 

:0. 

4.    .-. 

(52/+ 17)  (2/ -1)  = 

:0. 

6.    .-. 

y=  -  V,  or  1. 

6.    .-. 

x  =  3  +  22/=  - 

¥,  or  5. 

Check, 

,  for  X  =  - 

-  ¥,  y  =  -  V- 

-  ¥  +  ¥  =  ¥  = 

:3. 

¥/  +  -¥/  =  -%¥-  = 

:26. 

In  checking,  the  roots  must  be  properly  arranged  in  pairs. 
E.g.^  in  the  preceding  example 

X  =  —  J/  when  and  only  when  y  =  —  ¥, 
and  x  =  2  "       "       "         "     2/ =  5. 


284 


SIMULTANEOUS   QUADRATIC   EQUATIONS.  285 


EXERCISES.    CXXX. 

Solve  the  following  systems  of  equations : 


1.    xly  =  2. 
xy  =  ^. 

3.    X  +  2/  =  9. 
xy  =  45. 

5.    X  —  y  =  24:. 

xy  =  4212. 

7.    x  +  y  =  1.25. 
xy  =  0.375. 

9.    x^  +  y^  =  1274:. 
X  =  5y. 

11.    5(x  +  y)  =  xy. 
xy  =  180. 

13.    x-\-y  =  -6. 

xy  =  -2592. 

15.    ^x'  +  iy'  =  ll. 
^x-\-^y  =  5. 

17.    x^-\-xy  +  y^  =  63. 
x-y  =  -3. 

19.    (7  +  ^)  (6 +  2/)  =80. 
x  +  y  =  5. 

21.    a;2  +  7/2  =  500. 


3. 


x-y 


2.    x  +  y  =  100. 
x?/  =  2400. 

4.    X  —  y  =  11. 

6/ic  =  2//10. 

6.    tV^'  +  2/'  =  122. 
^a:;-3/  =  13. 

8.    2x^-{-y^- 100  =  0. 
jx-y-50  =  0. 

10.    3a;  +  4?/-8  =  10. 
x^  —  y^  =  —  5. 

12.    ia;2+ 1^/2-60  =  0. 
i  ^  +  i  2/  -  5  =  0. 

14.    14a^2_i22  7/2  =  100. 
x  =  Sy. 

16.    27  ic  + 33  2/ -60  =  0. 
Sx^-\-10y^-18  =  0. 

18.    0.01ic2  +  0.52/-2  =  0. 
0.1;z;-0.25y-3  =  0. 

20.    0.01 0^2  +  400?/ -25  =  0. 
0.5x  +  y-10  =  0. 

22.    a?  +  ?/  —  4  =  0. 
«      2/ 


286  ELEMENTS   OF   ALGE13RA. 

287.  2.  When  both  equations  are  quadratic.  In  this  case, 
X  can  be  found  in  terms  of  ?/  in  either  equation,  but,  in 
general,  the  value  will  involve  y'^.  In  this  case,  the  value 
of  X  substituted  in  the  other  equation  will  inyolye  2/*?  and 
hence  the  result  will  he  an  equation  of  the  fourth  degree. 

E.g.,  given  the  system  x^  —  y^  =  —  S. 

From  the  first  equation 

x=  ±  Vy2  _  3. 

Substituting  in  the  second, 

2(?/2-3)±3V2/2-3  +  7/  =  7. 
Isolating  the  radical,  squaring,  and  reducing,  we  have 
2  ?/  +  2  7/3  -  30  ?/2  -  13  ?/  -f  98  =  0, 
an  equation  of  the  fourth  degree. 

288.  Hence,  in  general,  two  simultaneous  quadratic  equa- 
tions involving  two  unknown  quantities  cannot  be  solved  hy 
means  of  quadratics. 

It  is  only  in  special  cases  that  such  systems  admit  of  solu- 
tion by  quadratics,  and  four  pairs  of  roots  should  aliv ays  be 
expected. 

A  few  of  the  more  common  of  these  special  cases  will 
now  be  considered. 

EXERCISES.    CXXXI. 

To  what  single  equations  of  the  fourth  degree  do  the 
following  systems  reduce  ? 

2.    y'^  -\-  2  X  —  xy  =  5. 

x"^  -\-  x  -\-  y  =  4:. 

4.    x'^  -\-  xy  -\-  y^  -\-  X  —  5  =  0. 
2x^  +  y^-x  +  y-3  =  0. 


1. 

x  +  y^  =  11. 

3. 

2x^-i-3x-t/  = 

:0. 

x'-32/  +  y  = 

:0. 

SIMULTANEOUS   QUADRATIC   EQUATIONS.  287 

289.  When  one  equation  is  homogeneous.  In  this  case  a 
)lution  is  always  possible.  For  if  ax^  +  hxy  +  cy^  =  0  is 
le  homogeneous  equation  we  can  divide  by  y^  and  have 

Qu  X  Of  "V 

—^  +  h \-  c  =  0,  a  quaxlratic  in  -  •     Hence,  -  can  be 

f        y  y  y 

mnd  and  x  will  then  be  known  as  a  multiple  of  y,  and 
this  value  can  then  be  substituted  in  the  other  equation. 
E.g.^  to  solve  the  system 

1.  x2  _  5  a;^/  +  2/2  =  0. 

2.  x2  +  3x-4y  +  4=0. 
2     5/x 


(-^o(^^)=o■  . 


X      \         ^        -,         y 

5.  .-.  -  =  -,  or  2,  and  x  =  -,  or  2y. 

»  2/      2  2 

V 
Substituting  x  =  -  in  eqiuation  2,  we  have 

6.  ^  +  ^_4j,  +  4  =  0. 

7.  .-.  ^2  _  10^ +  16  =  0. 

8.  .-.  ?/  =  2,  or  8,  and  .-.  x  =  ^  =  1,  or  4. 
Substituting  x  =  2y  in  equation  2  and  reducing,  we  have 

10.  .-.  y=  _;|,-t  jiVl5. 

11.  .-.  x  =  22/=  -^±^iVTb. 

12..-.  jc  =  l,  4,  -i  +  iiVIs,  -i-iiVl5, 

and  y  =  2,  8,  -i  +  izVl5,  -i-iiVi5, 

these  roots  being  taken  in  pairs  in  the  order  indicated. 

Check.  All  of  these  roots  check.  While  the  substitution  of  the 
complex  roots  takes  time  and  patience,  it  is  the  only  method  of  deter- 
mining the  correctness  of  the  solution. 


288 


ELEMENTS  OF  ALGEBRA. 


EXERCISES.    CXXXII. 

Solve  the  following  systems  of  equations  : 

1.         CC^    4-    2/2    _    Irj^y    _    Q^ 

x-\-y  =  a. 

2.  Zx'^^xy-y''  =  h. 

x^  —  2xy  +  y^  =  0. 

3.  Bx^  -\-4txy  —  y^  =  0. 

x^  +  X  +  y  =  5. 

4.  x^  -\-  xy  +  X  —  y  =  —  2. 

2x^  —  xy-y^  =  0. 

5.  x^  +  3xy  +  3x~y  =  2. 

x^  +  2xy-3y^  =  0. 

6.  x'-y^  +  x  +  y=is, 

86  (x''  + 7/^=  97  xy. 

7.  2x^  +  3xy  +  4.y  =  18. 

x^  +  4:xy  =  12  if. 

8.  3x^^-^xy^3x-y  =  3. 

x^  -\-  xy  =  0. 

9.  x^  +  4:x  +  3y  +  y''=  -2. 

x(x-}-2y)-15y^  =  0. 

10.  x(x  +  y)  +  y(y^x)  =  4:X7/. 

^{^  +  y)  +  y  +  x  =  24.. 

11.  x'^-3x-\-4.y  +  2xy  =  24.. 

x^-\-3xy  =  4.  tf. 

12.  147iz;2  +  196a:?/  +  57?/2  =  0. 

x''  +  2xy  +  33  =  0. 


SIMULTANEOUS   QUADRATIC   EQUATIONS.         289 


290.   When  both  equations  are  homogeneous  except  for  the 
ibsolute  terms.     In  this  case  a  solution  is  always  possible 
quadratics.     For  if 

a^x^  +  bixy  +  c^y^  =  d^, 
id  a^x^  +  b^XT/  +  Czi/  =  d^, 

re  can  multiply  both  members  of  the  first  by  d^,  and  of 
le  second  by  di,  and  subtract,  and 

(a^dz  —  a^di)  x^  +  (b^d^  —  b^di)  xij  +  (c^d^  —  CgC^i)  3/^  =  0. 

lis  may  now  be  treated  as  in  §  289. 

JB-gr.,  to  solve  the  system 

1.  x2  +  3x?/-2?/2  =  2. 

2.  2  x2  _  5  x?/  +  6  2/2  =  3. 

Multiplying  both  members  of  equation  1  by  .3,  and  of  equation  2  by 
^,  and  subtracting,  we  have : 

3.  x2  _  19  JC2/+ 18  7/2  =  0. 
This  equation  is  easily  reducible.     If  it  were  not,  we  should  divide 

by  2/2  and  proceed  as  in  §  289. 

4.  .-.  (a;  -18?/)(x-?/)  =  0. 

5.  .-.  X  =  18y,  or  y. 
Substituting  18?/  for  x  in  1,  we  have 

6.  324  2/2  +  54  2/2  -  2  ^2  =  2. 

7.  .-.  2/  =  ±  i  V^  -  ±  9V  ^^» 
and  X  =  18  2/         =  ±  /t  ^^• 

Substituting  ?/  for  x  in  1,  we  have 

8.  2/2  +  3  2/2  -  2  2/  =  2. 

9.  .-.  y  =  ±  1,  whence  x  =  ±  1. 
Check.     All  of  these  results  check. 

E.g.,  try  X  =  ±  57  ^^7,  2/  =  ±  9V  ^^7. 
Substituting  these  values  in  equation  1, 

H  +  f|-Tl8=2. 

Substituting  in  equation  2, 


290  ELEMENTS   OF  ALGEBRA. 

29J..    Since  §  §  289  and  290  depend  upon  finding  the  value 

of  -;  or  of  -?  we  can  also  solve  by  letting  -  =  v,  ot7/  =  vx, 
y  X  X 

then  finding  v. 

E.g.,  ill  the  preceding  example  we  had  the  system 

1.  x^  +  3  x^/  -  2  ?/2  =  2. 

2.  2  x2  -  5  X2/  +  6  ?/2  z=  3. 

Let  -  =  u,  OT  y  =  vx.     Then,  from  1,  we  have 


X 

3- 

x2  +  3  ux2  -  2  ^2x2  =  2. 

4.    /. 

' 

1  +  3  u  -  2  u2 

Similarly, 

from  2 

,  we  have 

5. 

2  x2  -  5  vx2  +  6  U2x2  =  3. 

6.    .-. 

^ 

2  -  5  u  +  6  u2 

Equating 

the  values  of  x^, 

7. 

2                           3 

l+3u-2u2      2-5w  +  6d2 

Reducing, 

8. 

18  v2  _  19  u  4-  1  =  0, 

or 

{18u-l)(t;  -1)  =  0. 

9.   .-. 

v  =  j\,  or  1. 

10.    .-. 

7/  =  vx  =  tVx,  or  X. 

This  is  substantially  the  same  as  step  5  of  the  preceding  solution 
(p.  289),  and  the  rest  of  the  work  is  as  given  there. 

In  the  same  way  we  may  let  -  =  v,  or  x  =  vy.     We  should  then 
have,  from  equation  1, 

1)22/2  -I-  3  vy"^  -  2  ^2  _  2. 


v2  +  3  u  -  2 

Q 

Similarly,  from  2,  y^  = 


2  v-2  _  5  u  +  6 
Equating  these  values  of  y^,  v  can  be  found  as  above. 


SIMULTANEOUS   QUADRATIC   EQUATIONS.  291 

EXERCISES.    CXXXIII. 

Solve  the  following  systems  of  equations  : 


1. 

x'^  +  2xy  =  39. 
xy  +  2y^  =  &b. 

2. 

x^  +  3xy  =  2. 
3y^  +  xy  =  l. 

3. 

x'^  +  Zxy  =  54. 
xy-{.4.y^  =  im 

4. 

2x^  +  3xy  =  21. 
xy  -{-y"^  =  4. 

5. 

7}i^x^  -\-  n^y^  =  q^. 
x'^      y' 

6. 

7a;2-5xy  =  18. 
2/'              2/ 

7. 

3xy  +  y^-lS  = 

0. 

8. 

x'^-xy  +  y^  =  21. 

4.x''-{-xy-7  = 

0. 

y''-2xy  =  -15 

p 


9.    x!^  +  xy  -\-7/  =  139.  10.    aic2  4-  ^,  (a;^  +  t/)  =  m. 

5  y^  —  4:xy  =  —  75.  mf"  -f  c? (a^^  +  ?/^)  =  /?-. 

11.    x^-2xy  ^y''  =  51.  12.    3  cc^  -  5  x^/ +  2  t/^  =  14. 

169  x^  +  2  2/'  =  177.  2x''-hxy  ^Zy''  =  Q>. 

13.   2ic2  +  2a;2/  +  2/2  =  73.       14.    32?/2_2£c?/-ll  =  0. 
ic2  +  ic?/  +  2/  =  74.  x2  +  42/--^  =  10. 

15.  3x2  +  13a;y  +  8?/2  =  162. 

x^  —  xy  +  y'^  =  7. 

16.  (3a;  +  ?/)(3  2/  +  a;)=384. 

(x  -  2/)  (x  +  7/)  =  40. 

17.  3x2  +  4x2/4-57/2-48  =  0. 

4  x^  4-  5  x?/  —  36  =  0. 

18.  2x2 +  3x7/ -37/2 +  124  =  0. 
7  x2  -  XT/  -  2/^  +  49  =  0. 


292  ELEMENTS  OF  ALGEBRA. 

292.  When  the  equations  are  symmetric  with  respect  to  the 
two  unknown  quantities.  In  this  case  a  solution  is  always 
possible  by  quadratics.  The  solution  is  accomplished  by 
letting  X  =  u  -\-  V,  and  y  =  u  —  v,  and  first  solving  for  u 
and  V. 

E.g.,  given  the  system 
1.  x^  +  Zxy  -\-y^  =  Al. 

.2.  x^  +  y^  +  X  +  y  =  32. 

Let  x  =  u  +  V  and  y  =  u  —  v.     Then,  by  substituting  in  1,  we  have 

3.  5  w2  _  u2  _  41^  or  v^  =  5u'^-  41. 
Substituting  in  2, 

4.  u^-\-v^  +  u  =  16. 
Substituting  here  the  value  of  v^  from  3, 

5.  Qu^  +  u-67  =  0, 
or                                     (6w  +  19)(tt-3)  =  0. 

6.  .-.  u=  - V-,  or  3. 
Substituting  this  value  of  m  in  3, 

7.  v=  ±1  V329,  or  ±  2. 

8.  .-.  X  =  w  +  V  = -,  6,  or  1,  four  values  as  we  should 

expect  (§  287).  ^ 

9.  Since  the  equations  are  symmetric  with  respect  to  x  and  y,  y 
must  have  the  same  values,  always  arranged  so  that  x  -{-  y  shall  equal 
2  u.     (Why  ?) 

„  -  19  +  V329     -  19  -  V329    ^   , 

10.  .-.for      x  = ,  ,5,1, 

6  '  6  '    '    '  . 

-  19  -  V329     -  19  +  V329    ,    , 
we  have  y  = , ,1,5 

D  O 

All  of  the  results  check. 

It  should  be  noticed  that  a  set  of  equations  like 
x-y  =  l,  a;2  +  2/2  z=  25, 
is  symmetric  with  respect  to  x  and  —  y.     Hence,  if 
x  =  3,  or  —  2,  y  =  2,  or  —  3. 


SIMULTANEOUS   QUADRATIC   EQUATIONS.  293 

EXERCISES.    CXXXIV. 

Solve  the  following  systems  of  equations  : 
1.    jc2  _^  2/^  =  41.  2.    x^  ^xy  ^-y^  =  19. 

X  —  y  —  \.  X  -\-  y  =  5. 

3.    x^-xy  +  y'^  =  3.  4.    x^ -\- y^ -\- 3  (x -{- y)  =  4:. 

x^-{-xy  +  y^  =  T.  Sx^  +  4,xy  +  Sy^  =  3. 

5.    x  +  Vxy  4-  2/  =  14.  6.    x^  —  2.5  xy -\- y'^  =  0. 

x'  +  xy-{-y'  =  84.  2(x-\-  yf  =  3.6  (x'  +  y'). 


1  ,  1  55  o 


-  +  -  =  7.  y^x       5 

X       y 

9.    ic  (ic  +  2/)  —  40  =  0. 

y(y  +  x)~60  =  0. 

10.    2ic2_|_^2/  +  2  2/'-79.58. 

a;2- 2  a:?/ 4- 2/' =  21.29. 


p.       293.   3.  When  equations  above  the  second  degree  are  involved. 

In  general,  such  systems   cannot  he  solved  by  quadratics, 
although  they  can  be  solved  in  special  cases. 

E.g.,  x3  +  x2?/ +  2/3  =  11. 

x-y  =  -\. 

Here  x  =  y  —  \;   hence, 

(y-l)3  +  (y-l)2  2/  +  2/'rz:ll, 

or  3  2/^  —  5  ?/2  +  4  ^  —  12  =  0,  a  cubic  equation. 

Now  a  cubic  equation  may  sometimes  be  solved  by  factoring,  as 

here,  for  this  reduces  (§  104)  to 
,^  (?/-2)(3  2/2  +  2/  +  6)  =  0, 

^^fcwhence  y  =  2,  or  \{-l  ±i  ^71), 


294  ELEMENTS  OF  ALGEfiRA. 

294.  If  the  equations  are  symmetric  with  respect  to  the 
unknown  quantities,  they  often  yield  to  the  method  given 
in  §  292. 

E.g.,  to  solve  the  system 

1.  x^-\-y^  =  91. 

2.  x  +  y  =  7. 
Let  x=:w  +  v,  y  =  u  —  V.     Then 

3.  2  M^  +  G  uv^  =  91,  from  1. 

4.  w  =  |,        "     2. 

5.  .-.  i|^  +  21  «2  ^  91,  and  V  =  ±  i 

6.  .:  X  =  u  +  V  =  4,  or  3,  and  .-.  y  =  S,  or  4,  by  symmetry. 
This  system  is  easily  solved  in  other  ways,  as  by  dividing  the  mem- 
bers of  1  by  the  members  of  2,  etc. 

EXERCISES.    CXXXV. 

Solve  the  following  systems  of  equations  : 
1.    x^-\-if  =  72.  2.    x^-^y^  =  97. 

X  -\-  y  =  6.  X  -\-  y  =  1. 

3.    ic*  +  7/4  =  337.  4.    x^-y^  =  219. 

X  —  y  :=!.  X  —  y  =  3. 

5.    x^ -\- y^  =  4:14,9.  6.    x"^  +  y^ -^  xy  (x  +  y)  =  15i. 

x  +  y  =  9.  '  cc«  +  2/'-3(a;'  +  2/')  =  50. 

x^      y^  ^y        ^x       2      . 

1-1  =  1  1      1_10 

X       y         '  X       y      xy 

I 4 

9.   Vic  +  2/  +  —j=  —  4  =  0. 

x"  +  y^      34 
x«/         15 


V 


SIMULTANEOUS   QUADRATIC   EQUATIONS.  295 


295.  Special  devices  will  frequently  suggest  themselves, 
but  it  is  not  worth  while  to  attempt  to  classify  them.  A 
few  are  given  in  the  following  illustrative  problems. 

1.  Solve  the  system 

1.  x^'if'  -\-  xy  —  ^^^. 

2.  a;2  +  ?/2  =  5. 
From  1  we  have 

3.  (xy  —  2)  {^y  +  3)  =  0,  whence  xy  =  2,  or  —  3. 

4.  Addmg  2  xy  =  4  or  —  6  to,  and  subtracting  it  from,  the  respec- 
tive members  of  2,  we  have 

5.  x2  4.  2  xy  +  2/2  =  9,  or  -  1. 
x2-2xy  +  2/2  =  1,    "      11. 

6.  .-.  X  +  y  =  ±  3,  or  ±  i, 

X  -  ?/  =  ±  1,  "  ±  ViT. 

Adding,  and  dividing  by  2, 

2  2 

i+  VIT  i-  VTT   -i  +  vn   -^-VTT 

-2,1,-1,-2,        ^        ,        2        '  2  '  2  ' 

On  account  of  symmetry,  y  must  have  the  same  values,  arranged  so 
as  to  satisfy  step  6.  

I-  vTT  i  +  Vii   -  i-  ViT   -i+  Vn 

.•.y_l,2,  -2,  -1,         ^        ,         ^        ,  2  '  2  ■ 

All  of  the  results  check. 

E.g.^  consider  the  last  ones, 

-i  -  Vn             -  i  +  vTi 
x  = ^ ,      y  = ^ 

Substituting  in  equation  1, 

/  i  _  viT  -  i  +  vny     t  -  Vn  -  i  +  vn   ^ 
V      2      '       2      ^  2       "       2 

::3  (_  3)2  +  (_  3)  _  6  =  9  -  3  -  6  =  0. 
Substituting  in  equation  2, 

/_i_ViTV    /-t  +  ViiV    io  +  2iVn     io-2iViT 


296  ELEMENTS   OF  ALGEBRA. 

2.  Solve  the  system 


1.   X  =  a  V£C  +  y. 

2.   y  =  b^x  +  y. 

Adding, 

3. 

x  +  y  =  {a-^b)-Vx  +  y,OT 

X  +  2/  -  (a  +  6)  Vx  +  2/  =  0,  or 

4. 

Vx  +  2/  ( Vx  +  y  -  a  +  6)  =  0. 

5.   .-. 

Vx  +  2/  =  0,  or  a  +  6. 

Substituting  in 

1  and  2, 

X  =  0,  or  a  (a  +  b). 

y  =  0,   "   6  (a  +  6). 

The  results  check. 

3.  Solve  the  system 

1.  ic*  +  icy  +  2/*  =  481. 

2.  £c^  +  iC2/  +  2/2  =  37.    - 

Factoring  1,  by  §  114, 

3.  (x2  +  xy  +  y2)  (a;2  -  xy  +  2/2)  _  431. 

4.  .-.  37  (x2  -xy  +  2/2)  =  481,  or 

x^-xy  +  y^  =  13. 
Subtracting  from  2, 

5.  2xy  =  24,  whence  xy  =  12. 
Adding  to  2,  and  subtracting  from  4, 

6.  x^  +  2xy  +  y'^  =  49. 
x2-2x2/  +  ?/2  =  1. 

7.  .-.  X  +  2/  =  ±  7. 

X  -  2/  =  ±  1. 

8.  .-.  x  =  4,    -4,  3,    -3,     y  =  3,    -3,  4,    -4. 

Graphs.  For  the  graphic  representation  of  quadratic 
equations,  and  for  the  discussion  of  the  number  of  roots 
of  simultaneous  quadratic  equations  with  two  unknown 
quantities,  see  Appendix  IX.  If  Appendix  VIII  has  been 
studied,  this  may  be  taken  at  this  point. 


SIMULTANEOUS   QUADRATIC   EQUATIONS.  297 

MISCELLANEOUS    EXERCISES.    CXXXVI. 

Solve  the  following  systems  of  equations : 
1.    x^  *+  y^  =  h.  2.    a;2  —  £C2/  +  /  =  124. 

X  -{-  y  =  a.  x^  —  y^  =  4:4c. 

3.    x^  +  y^  =  3x.  4:.    yVy  =  17  Vy  + Ax. 


X2  -{-  y^  =  X.  x^  =  4  V^/  +  17 


X. 


5.    x''  +  y^  =  25xY-  6-    ^'  +  2/'-a^-2/  =  l- 
12  cci/  =  1-  xy  =  1. 

7.    V^  +  V^  =  12.  8.    3(x2  +  2/2)  =  io(a;  +  2/). 
x^-}-2f  =  3026.  9  (ic^  +  2/0  =  34  (a;*  +  y^). 


9.    (ic2  +  a;^/  +  2/^  ^^^  4-  2/^  =  185. 
(x2  -  cc?/  +  y^)  V^M^  ^  65_ 

2/      dVx      81  _  Vic 


X         2/         ^2/ 

"^  +  3^^  = 
X               y 

xWy 

^  y 

^x. 

1.    Vx^  +  144  +  Vy^  +  144  . 

=  35. 
=  144. 

2.    Vic2  -  2/^  -  Vcc^ 

+  f  +  ^ 

=  0. 

Vx  +  y  —  Vcc  —  ?/  =  1.5. 

13.  X +  y —  2Vxy —  ■\/x-{-^=2. 

Vx  +  V^  =  7. 

14.  V^  +  V^  =  x  —  y  =  x  —  Vxy  +  ?/. 

15.  ic2_  g^^_^9^2_4^_j_12?/=i-4. 

ic^  -  2  ic?/  +  3  ^2  -  4  X  +  5  2/  =  53. 


298  ELEMENTS  OF  ALGEBRA. 

II.  THREE  OR  MORE  UNKNOWN  QUANTITIES. 

In  general,  three  simultaneous  quadratic  equations  involv- 
ing three  unknown  quantities  cannot  he  solved  by  quadratics. 

Many  special  cases,  however,  admit  of  such  solution. 

The  same  is  true  if  one  equation  is  linear  and  the  other 
two  are  quadratic,  or  if  one  is  of  a  degree  higher  than  2. 

If,  however,  two  are  linear  and  the  other  quadratic,  a 
solution  is  possible  by  quadratics,  as  in  illustrative  prob- 
lem 2  on  p.  299. 

Illustrative  problems.     1.  Solve  the  system 

1.  ^xy  =  2x  +  2y. 

2.  2yz  =  3y  +  2z, 

3.  4.ZX  =  ^z  —  3x. 

Dividing  both  members  of  1,2,  3,  by  xy,  yz,  zx^  respectively,  we 
have 


4. 

^-hl 

5. 

-    \4- 

6. 

4  =  5         _?• 

X              z 

Adding  5  and  6, 

7. 

6  =  '  +  '- 
X     y 

Eliminating  ?/,  with  4  and  7, 

8. 

3  =  - ,  whence  x 

X 

.■.y=:2,  z  =  3. 

Check. 

6  =  2  +  4, 

12  =  6  +  6, 

12  =  15  -  3. 

SIMULTANEOUS   QUADRATIC   EQUATIONS.         299 

2.  Solve  the  system 

1.  x  +  y  —  2z  =  —  9. 

2.  Sx  +  2y-\-z  =  9. 

3.  a;2  +  2/'  +  ^'  =  30. 

Eliminating  z  from  1  and  2, 

9  -  7  X 

4.  y  =  — - — • 

o 

Eliminating  y  from  1  and  2, 

27  -X 

5.  2  = • 


5 

Substituting 

4  and  5  in  3,  and  reducing, 

6. 

6x2 -12a; +  4  =  0,  or 

(x-2)(5x-2)  =  0. 

7.    .-. 

X  =  2,  or  f. 

2/  =  -  1,  or  H. 

z  =  5,  or  6^^. 

Check  for  the  second  set  of  values. 

l  +  H-10||=-9. 

1  +  f  f  +  5.V  =  9. 

A  +  iM  +  Hm  =  Hir~  =  30 

EXERCISES.    CXXXVII. 

Solve  the  following  systems  of  equations 


4:y^  =  9  xz. 

2. 

ic^  +  y  +  052/ =  19 

x^  =  S6yz. 

7/2  +  ^2  _^  7/^  =  37 

9z^  =  4:  xy. 

^2  _^  ic2  +  ^x  -  28 

x^  4-  y'     5 
xyz          6 

4. 

cc?/^        9 
x^y      2 

^2  +  a;2          5 

xijz         3 

^2/^    =2 
2/  +  ^ 

2/2  +  ^2^13 

xyz        18 

iC2/^          6 

^  +  ic       7 

300  ELEMENTS   OF  ALGEBRA. 

III.    PROBLEMS   INVOLVING  QUADRATICS. 

EXERCISES.    CXXXVIII. 

1.  The  difference  of  two  numbers  is  11,  the  sum  of  their 
squares  901.     What  are  the  numbers  ? 

2.  The  sum  of  two  numbers  is  30,  the  sum  of  their 
squares  458.     What  are  the  numbers  ? 

3.  Pind  two  numbers  whose  sum,  whose  product,  and 
the  difference  of  whose  squares  are  all  equal. 

4.  The  sum  of  the  squares  of  two  numbers  is  421,  the 
difference  of  the  squares  29.     What  are  the  numbers  ? 

5.  A  certain  fraction  equals  0.625,  and  the  product  of 
the  numerator  and  denominator  is  14,440.  Required  the 
fraction. 

6.  The  sum  of  the  areas  of  two  circles  is  24,640  sq.  in., 
and  the  sum  of  their  radii  is  112  in.  Required  the  lengths 
of  their  radii. 

7.  The  product  of  the  numbers  2cc3  and  4=7/6,  in  which 
X  and  y  stand  for  the  tens'  digit,  x  being  twice  y,  is  103,518. 
What  are  the  tens'  digits  ? 

8.  If  a  certain  two-figure  number,  the  sum  of  whose 
digits  is  11,  is  multiplied  by  the  units'  digit,  the  product  is 
296.     Required  the  number. 

9.  Three  successive  integers  are  so  related  that  the 
square  of  the  greatest  equals  the  sum  of  the  squares  of  the 
other  two.     Required  the  numbers. 

10.  Separate  the  number  102  into  three  parts  such  that 
the  product  of  the  first  and  third  shall  be  102  times  the 
second,  and  the  third  shall  be  |  of  the  first. 


i 


SIMULTANEOUS   QUADRATIC   EQUATIONS.  301 

11.  Two  cubes  have  together  the  volume  407  cu.  in.,  and 
e  sum  of  one  edge  of  the  one  and  one  of  the  other  is 

1  in.     Required  the  volume  of  each. 

12.  If  the  product  of  two  numbers  is  increased  by  their 
the  result  is  89 ;  if  the  product  is  diminished  by  their 

m,  the  result  is  51.     Required  the  numbers. 

13.  One  of  the  sides  forming  the  right  angle  of  a  right- 
gled  triangle  is  f  the  other,  and  the  area  of  the  triangle 
5082  sq.  in.     Required  the  lengths  of  the  sides. 

14.  There  are  two  numbers  such  that  the  product  of  the 
st  and  1  more  than  the  second  is  660,  and  the  product  of 
e  second  and  1  less  than  the  first  is  609.     What  are  the 

numbers  ? 

15.  A  sum  of  money  at  interest  for  5  yrs.  amounts  to 
$4600.  Had  the  rate  been  increased  1%  it  would  have 
amounted  to  $40  more  than  this  in  4  yrs.  Required  the 
capital  and  the  rate. 

16.  The  product  of  the  numbers  £cl7  and  2?/2,  in  which 
X  stands  for  the  hundreds'  digit  of  the  first  and  ij  for  the 
tens'  of  the  second,  and  in  which  y  =  x  -\-  3,  is  83,054. 
Required  the  values  of  x  and  y. 

17.  Find  a  two-figure  number  such  that  the  product  of 
the  two  digits  is  half  the  number,  and  such  that  the  dif- 
ference between  the  number  and  the  number  with  the  digits 
interchanged  is  |  of  the  product  of  the  two  digits. 

18.  In  going  1732.5  yds.  the  front  wheel  of  a  wagon 
makes  165  revolutions  more  than  the  rear  wheel;  but  if 
the  circumference  of  each  wheel  were  27  in.  more,  the  front 
wheel  would,  in  going  the  same  distance,  make  only  112 
revolutions  more  than  the  rear  one.  Required  the  circum- 
ference of  each  wheel. 


302  ELEMENTS   OF  ALGEBRA. 

19.  The  floor  of  a  certain  room  has  210  sq.  ft.,  each  of 
the  two  side  walls  135  sq.  ft.,  and  each  of  the  two  end  walls 
126  sq.  ft.     Eequired  the  dimensions  of  the  room. 

20.  A  certain  cloth  loses  ^  in  length  and  ^i^  in  width  by 
shrinking.  Eequired  the  length  and  width  of  a  piece  which 
loses  3.68  sq.  yds.,  and  which  has  its  perimeter  decreased 
3.4  yds.  by  shrinking. 

21.  A  rectangular  field  is  119  yds.  long  and  19  yds.  wide. 
How  much  must  the  width  be  decreased  and  the  length 
increased  in  order  that  the  area  shall  remain  the  same 
while  the  perimeter  is  decreased  24  yds.  ? 

22.  Two  points  move,  each  at  a  uniform  rate,  on  the  arms 
of  a  right  angle  toward  the  vertex,  from  two  points  50  in. 
and  136.5  in.,  respectively,  from  the  vertex.  After  7  sees. 
the  points  are  85  in.  apart,  and  after  9  sees,  they  are  68  in. 
apart.     Eequired  the  rate  of  each. 

23.  There  are  two  lines  such  that  if  they  are  made  the 
sides  of  a  right-angled  triangle  the  hypotenuse  is  17  in. ; 
but  if  one  be  made  the  hypotenuse  and  the  other  a  side,  the 
remaining  side  is  such  that  the  square  constructed  upon  it 
contains  161  sq.  in.     How  long  are  the  two  lines  ? 

24.  There  is  a  fraction  whose  numerator  being  increased 
by  2  and  denominator  diminished  by  2,  the  result  is  the 
reciprocal  of  the  fraction ;  but  if  the  denominator  is  in- 
creased by  2  and  the  numerator  diminished  by  2,  the  result 
is  ly^  less  than  the  reciprocal.     Eequired  the  fraction. 

25.  If  the  numerator  of  a  fraction  is  decreased  by  2,  and 
the  new  fraction  added  to  the  original  one,  the  sum  is  If ; 
if  the  denominator  is  decreased  by  2,  and  the  new  fraction 
added  to  the  original  one,  the  sum  is  2^l.  Eequired  the 
fraction. 


SIMULTANEOUS   QUADRATIC   EQUATIONS.  303 

REVIEW   EXERCISES.    CXXXIX. 

2.  Form  the  equation  whose  roots  are  0,  i,  i. 

3.  Solve  w" /a^  =  a%  b^-b^  =  (b^y,  d" I d"  =  l/c-». 

4.  Solve  X  -\-  y  =.  a  -^b,  x  /a  —  y  /b  =  a  /b  —  b  /  a. 

5.  Solve  ^y  +  ^z  =  11,  3^  +  6ic  =  9,  ^x-^y  =  4.. 

6.  Construct  an  integral  quadratic  function  of  x  such 
that  /(2)  =  0  and  /(3)  =  0. 

7.  Simplify 

\  {x""-^  ■  x^-'Y  •  (£c«  -i-  x'^Yl/  \  (cc«x^)«  -=-  (x^'  +  'Yl. 

Solve  the  following : 

8.  x^ -{- x^y^  +  y^  =  61. 

42 

x^  —  xi/  4-  y^  = 

-"       "^        xy 

9.  X  -{-  y  =  2xy  =  x^  —  y"^. 

10.    X -\- y  -\- (x -{-  y)^  -  12  =  0. 
x^  +  2/'  -  45  =  0. 

3 


I 


11.   ^-—4-^ —  . 

x-\-y      X  —  y      4 

2x3  +  6V  =  ^(^'-2/T 

12.  (3  cc  +  4  ?/)  (7  a;  -  2  2/)  +  3  a;  +  4  ?/  =  44. 
(3  X  +  4  2/)  (7  a;  -  2  2/)  -  7  X  +  2  2/  =  30. 

13.  17  (cc  +  7/)"-^  -  7 (x  +  yfx-^  =  l^x{x.-\-  y)"^. 

(x  -  y)^  =  y-l- 


CHAPTER   XVI. 

INEQUALITIES. 
MAXIMA  AND  MINIMA. 

296.  Having  given  two  real  and  unequal  numbers,  a  and 
h,  a  —  b  cannot  be  zero,  li  a  —  h  i^  positive,  a  is  said  to 
be  greater  than  b ;  if  negative,  a  is  said  to  be  less  than  b. 

E.g.,  3  > 2  because  3  —  2  is  positive, 

-2>-3"        -2-(-3)is  positive, 
•  -8<-2"        -3-(-2)is  negative. 
If  a  >  0,  then  a  is  positive,  and  if 

a<0,    "     "  "  negative. 

297.  The  inequalities  a>  b,  c>  d  are  called  inequalities 
in  the  same  sense,  and  similarly  for  a<b,  c< d.  But  a>b, 
G<d  are  called  inequalities  ih  the  opposite  sense,  and  similarly 
for  a  Kb,  G>d. 

298.  In  this  chapter  the  letters  used  to  represent  numbers 
will  be  understood  to  represent  positive  and  real  finite 
numbers,  except  as  the  minus  sign  indicates  a  negative 
number. 

299.  Just  as  we  distinguish  two  classes  of  equalities, 
(1)  equations  and  (2)  identities,  so  in  inequalities  we  have 
two  classes,  (1)  those  which  are  true  only  for  particular 
values  of  a  quantity  called  the  unknown  quantity,  and  (2) 
those  which  are  true  for  all  values  of  the  letters. 

JB.gr. ,  X  +  2  >  3  is  true  only  when  a  >  1,  but  a  +  &  >  &  is  always  true. 
304 


INEQUALITIES. 


305 


300.  If  a  variable  quantity,  x,  cannot  be  greater  than  a 
constant,  m,  but  can  equal  it  or  approach  indefinitely  near 
it  in  value,  then  m  is  called  the  maximum  value  of  x. 

Similarly,  if  ic  <  m  but  can  equal  it  or  approach  indefi- 
nitely near  it  in  value,  then  m  is  called  the  minimum  value 
of  X. 

E.g.,  (x  —  1)^<^0,  because  it  is  the  square  of  a  real  quan- 
tity and  hence  cannot  be  negative.  But  {x  —  iy  can  equal 
0  by  letting  a;  =  1.     Hence,  0  is  the  minimum  value  of 

(X  -  1)^. 

Since  we  shall  need  the  subject  of  inequalities  in  only  a 
few  cases  in  our  subsequent  work,  we  shall  present  but  a 
few  of  the  fundamental  theorems.  It  is  evident,  however, 
that  the  subject  is  an  extensive  one,  covering  simple  inequali- 
ties, quadratic  inequalities,  etc.,  together  with  simultaneous 
inequalities  corresponding  to  simultaneous  equations. 


301.    The  axioms  of  inequalities.     The  following  axioms 
have  already  been  assumed  and  used : 

Ax.  4.     If  equals  are  added  to  unequals,  the  sums  are 
unequal  in  the  same  sense. 

Ax.  5.     If  equals  are  subtracted  froin  unequals,  the  re- 
mainders are  unequal  in  the  same  sense. 


These  are  easily  demonstrated,  thus : 

1.  If  a  >  6,  then  a  —  his,  positive. 

2.  Then  a -'b  =  a -\-k -k -h 

=  {a  +  k)-{k-^h) 
and  this  expression  is  positive. 

3.  .-.  a  +  k>h-\-k. 
Similarly  for  ax.  5. 


§296 


§296 


Theorems.    Three  important  theorems  of  inequalities  will 
now  be  proved,  the  first  two  corresponding  to  axs.  6  and  8. 


306  ELEMENTS   OF  ALGEBRA. 

302.  Theorem.  If  unequals  are  multiplied  hy  equals,  the 
products  are  unequal  in  the  same  or  in  the  opposite  sense, 
according  as  the  multiplier  is  positive  or  negative. 

Proof.    1.  If         a>  h,  then  a  —  h  is  positive.  §  296 

2.  Then    k(a  —  b)  is  positive, 

and      —k(a  —  b)  is  negative.  §  296 

3.  .*.  ka  —  kh  is  positive, 

and       —  A;a —(— >t&)  is  negative. 

4.  .'.  ka^kb, 

and       ~ka<-  kb.  §  296 

In  this  discussion  the  multiplier  is  supposed  to  be  neither  zero  nor 
infinite. 

303.  Theorem.    i/'a>b,  then  a'">b'". 

Proof.    1.  a  —  b'\s  positive.  §  296 

2.  .-.  {a^-^  +  a'^-^  H \-  ab'''-^  +  b'^-'^)  {a  —  b) 

is  positive,  because  the  multiplier  is  evidently 
a  positive  quantity. 

3.  .".a"*  —  ^'"  is  positive,  because  this  is  the  prod- 
uct of  the  expressions. 

4.  .-.  a"'>^"'.  §  296 

304.  Theorem.    //  a  ^^  b,  a^  +  b^  >  2  ab. 

Proof.  1.  (<x  —  ^)^  >  0,  because  (a  —  by  is  positive,  being 
the  square  of  a  real  number.  It  is  not  0,  for 
a4^b. 

2.  .-.  a^-2ab  +  b''>0. 

3.  .-.  a''  +  b^>2ab. 
Evidently  a'^  +  6'^  =  2  a6,  if  a  =  6. 


INEQUALITIES.  307 

Illustrative  problems.     1.  Prove  that  ic^  >  2  x  —  1,  if  aj^tl. 
We  have  x2  +  1  >  2  x,  by  §  304. 

2.  x^'^*^  +  yP^'^> x^Y  -^  x'^yp,  if  x4^y. 

1.  This  is  true  if  xP  +  'i  —  xPyi  +  yp  +  'i  —  x^yP  is  positive. 

2.  Or  if  xP  (x9  —  yi)  —  yP  {xi  —  yi)  is  positive. 

3.  Or  if  {xP  —  yP)  {xi  —  yi)  is  positive. 

4.  But  both  factors  are  positive  if  x  >  ?/,  and  both  factors  are  nega- 
ive  if  X  <  ?/,  and  in  either  case  their  product  is  positive. 

3.  Which  is  greater,  2  +  V3,  or  2.5  +  V2  ? 


1. 

2  +  Vs  =  2.5  +  ^2,  according  as 

2. 

7  +  4V3I8I  +  5V2,  squaring. 

§303 

3. 

Or  as 

-  1^  +  4  V3  1  5  V2. 

Ax.  5 

4. 

Or  as 

49/g  -  10  Vs  1  50. 

§303 

5. 

Or  as 

-10V3|A- 

Ax.  6 

6.    But  a  negative  number  is  less  than  a  positive  one. 

.-.2  +  V3<2.5  +  V2. 

CC  1  1         iC 

4.  Solve  the  inequality  2ic  —  -  +  ->3aj  —  -  +  -• 

1.  12x-2x  +  3>18x-2  +  x.  §302 

2.  .-.  —  9x>-5.  Ax.  6 

3.  .-.  3;  <  f ,  and  f  is  the  maximum  value.   §  300 

Check.     If  X  =  f ,  the  inequality  becomes  an  equation.     If  x  >  f ,  the 
sense  of  the  inequality  is  reversed. 


5.  Solve  the  inequality  a-^  —  5  a?  +  6  <  0. 

1.  (x  —  2)  (x  —  3)  <  0,  and  hence  is  negative. 

2.  The  smaller  factor,  x  —  3,  is  negative,  and  the  other  positive. 

3.  .-.  x>2  and  x<3,  or  2<x<3. 


308  ELEMENTS  OF  ALGEBRA. 

6.  Show  that  the  minimum  value  of  x^  —  8  cc  +  22  is  6. 

1.  Let  x^  —  8x  -}-22  =  y,  in  which  we  have  to  find  the  minimum 
value  of  y. 

2.  Then  x^  -  8  x  +  22  -  ?/  =  0.  . 

3.  .-.  X  =  4  ±  Vy  —  6,  and  y  cannot  be  less  than  6  without  making 
X  complex. 

7.  Divide  the  number  6  into  two  parts  such  that  their 
product  shall  be  the  maximum. 

1.  Let  X  and  6  —  x  be  the  parts. 

2.  Then  x  (S  —  x)  =  ?/,  in  which  we  have  to  find  the  maximum  value 
oiy. 

3.  Solving  for  x,  x  =  3  ±  v  9  —  y,  and  y  cannot  be  greater  than  9 
without  making  x  complex. 

4.  When  y  =  9,  x  =  3 ;   .-.  the  parts  are  3  and  3. 

Check.     3-3  =  9;   but  2  (6  -  2)  =  2  •  4  =  8,  a  smaller  number. 

EXERCISES.     CXL. 

1.  What  is  the  nature  of  the  inequality  resulting  from 
subtracting  unequals  from  equals  ?     Prove  it. 

2.  Investigate  the  addition  of  unequals  to  unequals. 

3.  Also  the  subtraction  of  unequals  from  unequals. 

4.  Show  that  the  maximum  value  of  4  a;  —  a:;^  is  4,  and 
that  2  is  the  value  of  x  which  makes  this  f(x)  a  maximum. 

5.  If  f(x)  =  x^  -{-  X  -\- 1,  show  that  x  =  —  0.5  renders 
f(x)  a  minimum,  and  find  the  minimum. 

6.  Prove  that,  in  general,  x^  -\-  l>x^  -\-  x.      What  is 
the  exception  ? 

7.  Also  that  (x  +  yy>4:  xy. 

8.  Solve  the  inequality  x'^  -\-hx'>  —  ^. 


INEQUALITIES.  309 

9.    Prove  that  {a  +  h){b  -\-  c)  (c  +  a)>%  abc. 

10.  Prove  that  the  mininmrn  value  of  a;^  — 10  cc  +  35  is  10. 

11.  Solve  the  inequality  5xH-2>3cc  +  -  —  7.  Check 
[the  result. 

X  —  3 

12.  Solve  the  inequality >  0.     Check. 

13.  Required  the  length  of  the  sides  of  the  maximum 
rectangle  of  perimeter  16. 

14.  Prove  that  if  the  sum  of  two  factors  is  k,  a  constant, 
^he  maximum  value  of  their  product  is  k^/A. 

15.  Show  that  if  a  square  is  inscribed  in  a  square  whose 
area  is  16,  its  corners  lying  on  the  sides  of  the  larger  square, 
its  area  <|:  8. 

16.  If  a,  h,  G  are  three  numbers  such  that  any  two  are 
together  greater  than  the  third,  then 

a''  +  h^  +  c^  <2  ah  +  2hc  +  2  ca. 

17.  Solve  the  inequality  x^  —  3x<  10. 

18.  Solve  the  inequality  x(x,  —  10)  <  11. 

19.  Find  the  maximum  value  of  8  cc  —  x^,  and  also  the 
value  of  X  that  renders  this  f{x)  a  maximum. 

20.  Find  the  minimum  value  of  x(x  -\- 10),  and  also  the 
value  of  X  that  renders  this  f(x)  a  minimum. 

21.  Required  the  area  of  the  largest  rectangle  having  the 
perimeter  20  inches.     How  do  the  sides  compare  in  length  ? 

22.  Required  the  area  of  the  largest  rectangle  having  the 
perimeter  p  inches.     How  do  the  sides  compare  in  length  ? 


CHAPTER   XVII. 

RATIO,  VARIATION,  PROPORTION. 

I.     RATIO. 

305.  The  ratio  of  one  number,  a,  to  another  number,  h,  of 
the  same  kind,  is  the  quotient  -  • 

S2  2 

Thus,  the  ratio  of  $  2  to  $5  is  — ,  or  -,  or  0.4,  but  there  is  no 

$5  5 

ratio  of  $2  to  5  ft.,  or  $10  to  2.     Here,  as  elsewhere  in  algebra,  how- 
ever, the  letters  are  understood  to  represent  pure  (abstract)  number. 

A  ratio  may  be  expressed  by  any  symbol  of  division,  e.g., 
by  the  fractional  form,  by  ^,  by  /,  or  by  : ;  but  the  symbols 

generally  used  are  the  fraction  and  the  colon,  as  y?  or  a:b. 

306.  In  the  ratio  a:h,  a  \b  called  the  antecedent  and  h  the 
consequent. 

307.  The  ratio  h:a  \&  called  the  inverse  of  the  ratio  a  :  b. 

308.  If  two  variable  quantities,  x,  y,  have  a  constant 
ratio,  r,  one  is  said  to  vary  as  the  other. 

E.g.,  the  ratio  of  any  circumference  to  its  diameter  is  ;r  =  3.14159; 
hence,  a  circumference  is  said  to  vary  as  its  diameter. 

X 

If  -  =  r,  then  x  =  ry.     The  expression  "  x  varies  as  y  " 
is  sometimes  written  xccy,  meaning  that  x  —  ry. 
li  X  =  r  '  -■>  ic  is  said  to  vary  inversely  as  y. 
310 


If  RATIO,    VARIATION,    PROPORTION.  311 

309.  If  two  variable  quantities,  x,  ij,  have  the  same  ratio 
as  two  other  variable  quantities,  x',  y\  then  x  and  y  are  said 
to  vary  as  x'  and  y'.  And  if  any  two  values  of  one  variable 
quantity  have  the  same  ratio  as  the  corresponding  values 
of  another  variable  quantity  which  depends  on  the  first, 
then  one  of  these  quantities  is  said  to  vary  as  the  other. 

E.g.,  the  circumference  c  and  diameter  d  of  one  circle  have  the 
same  ratio  as  the  circumference  c'  and  diameter  d'  of  any  other  circle ; 
hence,  c  and  d  are  said  to  vary  as  c'  and  d\ 

If  two  rectangles  have  the  same  altitude,  their  areas  depend  on 
tlieir  bases;  and  since  any  two  values  of  their  bases  have  the  same 
ratio  as  tlie  corresponding  values  of  their  areas,  their  areas  are  said  to 
vary  as  their  bases. 

310.  Applications  in  geometry.  Similar  figures  may  be 
described  as  figures  having  the  same  shape,  such  as  lines, 
squares,  triangles  whose  angles  are  respectively  equal, 
circles,  cubes,  or  spheres.  It  is  proved  in  geometry  that 
in  two  similar  figures 

1.  Any  two  corresponding  lines  vary  as  any  other  two 
correspo7iding  lines. 

2.  Corres20O7iding  areas  vary  as  the  squares  of  any  two 
corresponding  lines. 

3.  Corresponding  volumes  vary  as  the  cubes  of  any  two 
corresponding  lines. 

E.g.,  in  the  case  of  two  spheres,  the  circumferences  vary  as  the 
radii,  the  surfaces  vary  as  the  squares  of  the  radii,  the  volumes  vary 
as  the  cubes  of  the  radii. 

These  facts  are  easily  proved.  Let  s,  s'  stand  for  the  surfaces  of 
two  spheres  of  radii  r,  r',  respectively.  Then  we  know  from  mensu- 
ration that 

8  =  4  7irr"2,  and  s'  =  4  7rr'2, 

s  _  4  Ttr"^  _  r^ 

Hence,  the  surfaces  vary  as  the  squares  of  the  radii.     In  like  manner 
the  volumes  might  be  considered. 


312  ELEMENTS  OF   ALGEBRA. 

Illustrative  problems,  1.  If  the  ratio  of  x^  to  3  is  27,  find 
the  value  of  x. 

•.•  —  =  27,  .-.  x2  =  3  •  27  =  81,  .-.  X  =  ±  9,  and  each  value  checks. 

o 

2.  If  a  sphere  of  iron  weighs  20  lbs.,  find  the  weight  of 
a  sphere  of  iron  of  twice  the  surface. 

1,  Let  ri,  ri  be  the  respective  radii. 

2.  Then  4  itr-^  =  i  •  4  Ttr<^^  because  the  surface  of  a  sphere  =  4  itr'^. 
(p.  311.) 

3.  ...  *:?=V2. 

4.  And  •••  the  volumes  (and  hence  the  weights)  vary  as  the  cubes  of 
the  radii  (§  310),  and  •.•  *^  =  (V2)3  =  2  V2. 

5.  .'.  the  second  sphere  weighs  2  v2  times  as  much  as  the  first. 

2  V^  •  20  lbs.  =  56.57  lbs.,  nearly. 

EXERCISES.    CXLI. 

1.  The  ratio  of  625  to  x^  is  5.     Find  x. 

2.  Find  X  in  the  following  ratios  : 

(a)  4.:x^=9.  (b)  ^c^ :  27  =  300.  (c)  cc  =  y^^  :  x. 

,  t^      X  _  .    .       0\) 

3.  Find  X  in  the  following  ratios : 

(a)  -==2.4.  (b)-s  =  y  («)  13^  =  49  • 

(d)  7:x  =  4.9.  (e)  (r^ :  5  =  V- 

4.  One  cube  is  1.2  times  as  high  as  another.  Find  the 
ratio  of  (1)  their  surfaces,  (2)  their  volumes. 

5.  The  surfaces  of  a  certain  sphere  and  a  certain  cube 
have  the  same  area.  Find,  to  0.01,  the  ratio  of  their  vol- 
umes. 


I   .„...__  ™ 

^H^  311.  Applications  in  business.  Of  the  numerous  applica- 
^^ftions  of  ratio  in  business,  only  a  few  can  be  mentioned,  and 
f^^ot  all  of  these  commonly  make  use  of  the  word  "  ratio." 

In  computing  interest,  the  simple  interest  varies  as  the 
time,  if  the  rate  is  constant ;  as  the  rate,  if  the  time  is  con- 
stant ;  as  the  product  of  the  rate  and  the .  number  repre- 
senting the  time  in  years  (if  the  rate  is  by  the  year),  if 
neither  is  constant. 

I.e.,  for  twice  the  rate,  the  interest  is  twice  as  much,  if  the  time  is 
constant ;  for  twice  the  time,  the  interest  is  twice  as  much,  if  the  rate 
is  constant ;  but  for  twice  the  time  and  1.5  times  the  rate,  the  interest 
is  2  •  1.5  times  as  much. 

The  common  expressions  "  2  out  of  3,"  "  2  to  5,"  "  6  per 
cent "  (merely  6  out  of  100)  are  only  other  methods  of  stat- 
ing the  following  ratios  of  a  part  to  a  whole,  f ,  f ,  yf ^,  or 
the  following  ratios  of  the  two  parts,  ^,  f ,  -^\. 

E.g.,  to  divide  $100  between  A  and  B  so  that  A  shall  receive  |2  out 
of  every  $3,  is  to  divide  it  into  two  parts 

(1)  having  the  ratio  2  : 1,  or 

(2)  so  that  A's  share  shall  have  to  the  whole  the  ratio  2  :  3,  or 

(3)  so  that  B's  share  shall  have  to  the  whole  the  ratio  1  :  3. 

EXERCISES.    CXLII. 

1.  Divide  $1000  so  that  A  shall  have  $7  out  of  every  |8. 

2.  Divide  $500  between  A  aHd  B  so  that  A  shall  have 
$0.25  as  often  as  B  has  $1.25. 

3.  The  area  of  the  United  States  is  3,501,000  sq.  mi., 
and  the  area  of  Russia  is  8,644,100  sq.  mi.  Express  the 
ratio  of  the  former  to  the  latter,  correct  to  0.01. 

4.  The  white  population  of  the  United  States  in  1780 
was  2,383,000 ;  in  1790,  3,177,257 ;  in  1880,  43,402,970 ; 
in  1890,  54,983,890.  What  is  the  ratio  of  the  population 
in  1790  to  that  in  1780  ?    in  1890  to  that  in  1880  ? 


314  ELEMENTS  OF  ALGEBRA. 

312.  Applications  in  physics,  (a)  Specific  gravity.  The 
specific  gravity  of  any  substance  is  the  ratio  of  the  weight 
of  that  substance  to  the  weight  of  an  equal  volume  of  some 
other  substance  taken  as  a  standard. 

In  the  case  of  solids  and  Uquids,  distilled  water  is  usually  taken  as 
the  standard.  Thus,  the  specific  gravity  of  mercury,  of  which  1  1 
weighs  13.596  kg,  is  13.596,  because  a  liter  of  water  weighs  1  kg,  and 

13.596  kg  :1  kg  =  13.596. 

In  the  case  of  gases  either  hydrogen  or  air  is  usually  the  standard. 

The  following  table  will  be  needed  for  reference : 

Specific  Gravities. 
Mercury,  13.596.  Silver,    10.5. 

Nickel,        8.9.  Gold,     19.3. 

Weights  of  Certain  Substances. 

1  1  of  water,  1  kg.  1  cm'  of  water,  1  g. 

1  cu.  ft.  of  water,  about  62. 5  lbs. 

Example.     What  is  the  weight  of  1  cu.  in.  of  copper  ? 

1.  1  cu.  ft.  of  water  weighs  62.5  lbs. 

2.  .-.  1  cu.  in.  of  water  weighs  62.5  lbs  -=-  1728. 

3.  .-.  1  cu.  in.  of  copper  weighs  8.9  •  62.5  lbs.  -f-  1728,  or  5.15  oz. 

EXERCISES.    CXLIII. 

1.  What  is  the  weiglit  of  a  cubic  foot  of  gold  ? 

2.  What  is  the  weight  of  1  cm^  of  nickel  ?    of  silver  ? 

3.  The  specific  gravity  of  ice  is  0.92,  of  sea-water  1.025. 
To  what  depth  will  a  cubic  foot  of  ice  sink  in  sea-water  ? 

4.  From  ex.  3,  how  much  of  an  iceberg  500  ft.  high  would 
show  above  water,  the  cross-section  being  supposed  to  liave 
a  constant  area  ? 


v-" 

1 

A 

F 

A  1 
P 

f 

^ 

P 

A' 

P' 

B~ 

w' 

^ 

1 
w 

P 

RATIO,    VARIATION,    PROPORTION.  315 

313.  (b)  Law  of  levers.  If  a  bar,  ^^,  rests  on  a  fulcrum, 
^F,  and  has  a  weight,  w,  at  . 

f.^,  then  by  exerting  enough 
[pressure,  p,  at  A  the  weight 
,can  be  raised.  In  the  first 
igure  the  pressure  is  down- 
ward (positive  pressure) ;  in 
^he  second  it  is  upward  (neg- 
itive  pressure). 

There  is  a  law  in  physics  that,  if  p',  tv'  represent  the 
lumber  of  units  of  distance  AF,  FB,  respectively,  and  p, 
the  number  of  imits  of  pressure  and  weight,  respectively, 
shen  Pp[_. 

ww' 

In  the  first  figure  p,  to,  p%  w'  are  all  considered  as  positive  ;  in  the 
jcond  figure  p  is  considered  as  negative  because  the  pressure  is  up- 
i^ard,  and  \o'  is  considered  as  negative  because  it  extends  the  other 
my  from  F.     Hence,  the  ratio  pp' :  low'  =  1  in  both  cases. 

Example.  Suppose  AF  =  25  in.,  FB  =  14  in.,  in  the 
first  figure.  What  pressure  must  be  applied  at  A  to  raise 
a  weight  of  30  lbs.  at  5  ? 

25  w 

1.  By  the  law  of  levers  =—  =  1. 

^  14-30 

2.  .-.  p  = ^  =  10.8,  and  .-.  the  pressure  must  be  16.8  lbs. 

25 


EXERCISES.    CXLIV. 

1.  Two  bodies  weighing  20  lbs.  and  4  lbs.  balance  at  the 
ends  of  a  lever  2  ft.  long.     Find  the  position  of  the  fulcrum. 

2.  The  radii  of  a  wheel  and  axle  are  respectively  4  ft. 
[and  6  in.  What  force  will  just  raise  a  mass  of  56  lbs.,  fric- 
ition  not  considered  ? 


B16  ELEMENTS  OE  ALGEBRA. 


REVIEW  EXERCISES.  CXLV. 

1.  In  each  figure  on  p.  315,  what  must  be  the  distance 
AF  in  order  that  a  pressure  of  1  kg  may  raise  a  weight  of        . 
100  kg  3  dm  from  i^'?  I 

2.  If  a  sphere  of  lead  weighs  4  lbs.,  find  the  weight  of  a 
sphere  of  lead  of  (1)  twice  the  volume,  (2)  twice  the  sur- 
face, (3)  twice  the  radius.  ' 

3.  A  nugget  of  gold  mixed  with  quartz  weighs  0.5  kg ; 
the  specific  gravity  of  the  nugget  is  ^.h^  and  of  quartz  2.15. 
How  many  grams  of  gold  in  the  nugget  ? 

4.  A  vessel  containing  1 1  and  weighing  0.5  kg  is  filled 
with  mercury  and  water ;  it  then  weighs,  with  its  contents, 
3  kg.     How  many  cm^  of  each  in  the  vessel  ? 

5.  What  pressure  must  be  exerted  at  the  edge  of  a  door 
to  counteract  an  opposite  pressure  of  100  lbs.  halfway  from 
the  hinge  to  the  edge  ?  one-third  of  the  way  from  the  hinge 
to  the  edge  ? 

6.  Explain  Newton's  definition  of  number :  Number  is 
the  abstract  ratio  of  one  quantity  to  another  of  the  same 
kind.  What  kinds  of  numbers  are  represented  in  the  fol- 
lowing cases  :  5  ft. :  1  ft.,  1  ft.  :  5  ft.,  the  diagonal  to  the 
side  of  a  square,  the  circumference  to  the  diameter  of  a 
circle  ? 

7.  The  depths  of  three  artesian  wells  are  as  follows : 
A  220  m,  B  395  m,  C  543  m;  the  temperatures  of  the 
water  from  these  depths  are:  A  19.75°  C,  B  25.33°  C, 
C  30.50°  C.  Rom  these  observations,  is  it  correct  to  say 
that  the  increase  of  temperature  is  proportional  to  the 
increase  of  depth  ?  If  not,  what  should  be  the  tempera- 
ture at  C  to  have  this  law  hold  ? 


RATIO,    VARIATION,   PROPORTION.  317 

The  Theory  of  Ratio. 

314.  A  ratio  is  called  a  ratio  of  greater  inequality,  of 
equality,  or  of  less  inequality,  according  as  the  antecedent 
is  greater  than,  equal  to,  or  less  than  the  consequent. 

315.  Theorem.  A  ratio  of  greater  inequality  is  dimin- 
ished, a  ratio  of  equality  is  unchanged  in  value,  and  a 
ratio  of  less  inequality  is  increased  by  adding  any  positive 
quantity  to  both  terms. 

Given         the  ratio  a  :  b,  and  p  any  positive  quantity. 


J.KI  yi 

uvc 

b+p  >  b 

clUOUi  Uillg 

«/»   u-  , 

<  ^' 

Proof 

1. 

a  -\-p  <  a 
b+p>  b 

according 

as 

ab  +  pb  ^  ab  -\-  ap. 

§  302 

ax.  6 

2. 

Or  as       2)h  =  ap,  or  as  ^  = 

a. 

§301 

3. 

I.e.,  as       a^b. 

316. 

q              n 

Theorem.     //  r- = - 

•^    b       d 

e 

~f           ' 

then 

each  of 

these 

,    a  +  c  H-e  +  •• 
ratios  equals  ^_^^_^^_^^^ 

Proof 

1. 

Let  T  =  k-     Then  k  =  -^  = 
b                                a 

e 
f~ 

•••. 

2. 

•*• 

a  =  kb, 
c  =  kd, 
e  =  kf--- 

3. 

.'.  a-\-  c  -\-  e  +  ■ 

•  ^  =  k{b  + 

d+f +■■■). 

Ax.  2 

4. 

a  -\-  c  -{-  e  -\-  • 
'  '  b  +  d-\-f-\-- 

c 

e 

Ax.  7 

318  ELEMENTS   OF  ALGEBRA. 

EXERCISES.    CXLVI. 

1.  Prove  that  the  product  of  two  ratios  of  greater  in- 
equality is  greater  than  either. 

2.  Consider  ex.  1  for  two  ratios  of  equality ;  of  less 
inequality.     Then  state  the  general  theorem  and  prove  it. 

3.  Find  the  value  of  x,  knowing  that  if  x  is  subtracted 
from  both  terms  of  the  ratio  ^  the  ratio  is  squared. 

4.  Is  the  value  of  a  ratio  changed  by  raising  both 
terms  to  the  same  power  ?  State  the  general  theorem 
and  prove  it. 

5.  Prove  (or  show  that  it  has  been  proved)  that  the 
value  of  a  ratio  is  not  changed  by  multiplying  both  terms 
by  the  same  number. 

6.  As  in  §  315,  consider  the  effect  of  subtracting  from 
both  terms  of  a  ratio  any  positive  number  not  greater  than 
the  less  term.     State  the  theorem  and  prove  it. 

a  -^  5  b  a  +  6b  ^ 


7.    Which  is  the  greater  ratio, 


a-{-  6b  a  +  7b 


8.  Which  is  the  greater  ratio,  —^j  or ~  ? 

y  —  2x  3y  —  2x 

«     TTTi,  •■u-o.t-  i  J.-       a  -\-  b  -\-  G  a  —  b  -\-  c  ^ 

9.  Which  IS  the  greater  ratio,  ; — —  ?  or  ; ? 

a  —  b  —  c  a  -i-b  —  c 

.r.T£^       ^       ^  ^-y    ^  a^  -\- b"^  +  c^       ab  -\-bc  +  cd 

10.  It  7  =  -  =  -)  prove  that  — ; ; — —  • 

b       G       d    ^  ab-\-bG-\-cd       b'^  +  c^ -\- d^ 

-.-.     ^f  ^      ^       ^       1   '  4.-u^7       3a4-5c  —  6e 

11.  It  J  =  -  =  -  =  k,  prove  that  k  = 


b      d     f      "-'^^  ^"-^      3b  +  5d-6f 

a 

b 


12.    If  -  >  -)  the  letters  standing  for  positive  numbers, 


prove  that  ->V^i:p^>^ 


RATIO,    VARIATION,    PROPORTION.  319 


II.     VARIATION. 

317.  It  has  already  been  stated  (§  308)  that  the  expres- 
lons  ^^x  varies  as  y,"  ^' x  varies  inversely  as  y,"  simply 

lean  that  the  ratios  x:y,  x\--)  are  respectively  equal  to 

)ine  constant.     These  are  merely  special  cases  of  x  =/(?/); 
)Y  x:y  —  k  reduces  to  a:;  =  ky,  whence  ic  is  a  function  of  y\ 

1  k 

milarly  x\-  —  k  reduces  to  a;  =  ->  whence  cc  is  a  function 
■     y  y 

Although  there  is  nothing  in  the  theory  of  variation  which 
not  substantially  included  in  the  theory  of  ratio,  the 
)hraseology  and  notation  of  the  subject  are  so  often  used 
■  in  physics  as  to  require  some  further  attention. 

Two  illustrations  from  physics  will  be  given  in  this  con- 
nection, the  one  relating  to  the  pressure  of  gases  and  the 
other  to  electricity.  While  neither  requires  much  algebra 
for  its  consideration,  each  offers  an  excellent  illustration  of 
the  use  of  variation  in  physics.  No  preliminary  knowledge 
of  physics  is  necessary,  however,  to  the  work  here  given. 

318.  Boyle's  law  for  the  pressure  of  gases.  It  is  proved  in 
physics  that  if  j^  is  the  number  of  units  of  pressure  of  a 
given  quantity  of  gas,  and  v  is  the  number  of  units  of 
volume,  then  p  varies  inversely  as  v  when  the  temperature 
remains  constant. 

This  law  was  discovered  in  the  seventeenth  century  by  Robert  Boyle. 
E.g.^  if  the  volume  of  a  gas  is  10  dm^  under  the  ordinary  pressure 
of  the  atmosphere  ("  under  a  pressure  of  one  atmosphere  "),  it  is 

-  as  much  when  the  pressure  is  2  times  as  great, 

i  u       u  u        u  u  u  ^  u 

n  ^ 

n  times  "      "    '     "       "  "        "  -  " 

n 


the  temperature  always  being  considered  constant. 


320  ELEMENTS   OF   ALGEBRA. 

Example.  A  toy  balloon  contains  3 1  of  gas  when  exposed 
to  a  pressure  of  1  atmosphere.  What  is  its  volume  when 
the  pressure  is  increased  to  4  atmospheres  ?  decreased  to  -^ 
of  an  atmosphere  ? 

1.  •.•  the  volume  varies  inversely  as  the  pressure,  it  is  ^  as  much 
when  the  pressure  is  4  times  as  great. 

2.  Similarly,  it  is  8  times  as  much  when  the  pressure  is  i  as  great. 

3.  .-.  the  volumes  are  0.75  1  and  24  1. 

EXERCISES.     CXLVII. 

1.  If  a  cylinder  of  gas  under  a  certain  pressure  has  its 
volume  increased  from  20  1  to  25  1,  what  is  the  ratio  of  the 
pressures  ? 

2.  A  certain  gas  has  a  volume  of  1200  cm^  under  a  pres- 
sure of  1033  g  to  1  cm^.  rind  the  volume  when  the  pres- 
sure is  1250  g. 

3.  A  cubic  foot  of  air  weighs  570  gr.  at  a  pressure  of  15 
lbs.  to  the  square  inch.  What  will  a  cubic  foot  weigh  at  a 
pressure  of  10  lbs.  ? 

4.  Equal  quantities  of  air  are  on  opposite  sides  of  a 
piston  in  a  cylinder  that  is  12  in.  long ;  if  the  piston  moves 
3  in.  from  the  center,  find  the  ratio  of  the  pressures.  Draw 
the  figure. 

5.  A  liter  of  air  under  ordinary  pressure  weighs  1.293  g 
when  the  barometer  stands  at  76  cm.  Find  the  weight  when 
the  barometer  stands  at  82  cm,  the  weight  varying  as  the 
height  of  the  barometer. 

6.  If  the  volume  of  a  gas  varies  inversely  as  the  height 
of  the  mercury  in  a  barometer,  and  if  a  certain  mass  occu- 
pies 23  cu.  in.  when  the  barometer  indicates  29,3  in.,  what 
will  it  occupy  when  the  barometer  indicates  30.7  in.  ? 


RATIO,    VAKIATION,    PROPORTION. 


321 


1319.    Problems  in  electricity.     The  great  advance  in  elec- 
■icity  in  recent  years  renders  necessary  a  knowledge  of 
!         such  technical  terms  as  are  in  everyday  use. 


When  water  flows  through  a 
pipe  some  resistance  is  offered 
due  to  friction  or  other  impedi- 
ment to  the  flow  of  the  water. 


A  certain  quantity  of  water 
flows  through  the  pipe  in  a  second, 
and  tliis  may  be  stated  in  gallons 
or  cubic  inches,  etc. 


A  certain  pressure  is  necessary 
to  force  the  water  through  tlie 
l)ipe.  This  pressure  may  be  meas- 
ured in  pounds  per  sq.  in.,  kilo- 
grams per  cm2,  etc. 


Hence,  in  considering  the  water 
necessary  to  do  a  certain  amount 
of  work  (as  to  turn  a  water-wheel) 
it  is  necessary  to  consider  not 
merely  the  pressure,  for  a  little 
water  may  come  from  a  great 
height,  nor  merely  the  volume, 
nor  merely  the  resistance  of  the 
pipe ;  all  three  must  be  consid- 
ered. 


When  electricity  flows  through 
a  wire  some  resistance  is  offered. 
This  resistance  is  measured  in 
ohms.  An  ohm  is  the  resistance 
offered  by  a  column  of  mercury 
1  mm2  in  cross-section,  106  cm 
long,  at  0°C. 

A  certain  quantity  of  electric- 
ity flows  through  the  wire.  This 
quantity  is  measured  in  amperes. 
An  ampere  is  the  current  neces- 
sary to  deposit  0.001118  g  of  silver 
a  second  in  passing  through  a  cer- 
tain solution  of  nitrate  of  silver. 

A  certain  ^^t'essure  is  necessary 
to  force  the  electricity  through  the 
wire.  This  pressure  is  measured 
in  volts.  A  volt  is  the  pressure 
necessary  to  force  1  ampere 
through  1  ohm  of  resistance. 

Hence,  in  considering  the  elec- 
tricity necessary  to  do  certain 
work  it  is  necessary  to  consider 
not  merely  the  voltage,  for  a  little 
electricity  may  come  with  a  high 
pressure,  nor  merely  the  amper- 
age, nor  merely  the  number  of 
ohms  of  resistance ;  all  three  must 
be  considered. 


The  names  of  the  electrical  units  mentioned  come  from  the  names 
of  three  eminent  electricians,  Ohm,  Ampere,  and  Volta. 


322  ELEMENTS  OF  ALGEBKA. 

320.  It  is  proved  in  physics  that  the  resistance  of  a  wire 
varies  directly  as  its  length  and  inversely  as  the  area  of  its 
cross-section. 

That  is,  if  a  mile  of  a  certain  wire  has  a  resistance  of  3.58  ohms, 
2  mi.  of  that  wire  will  have  a  resistance  of  2  •  .3.58  ohms,  or  7.16  ohms. 
Also,  1  mi.  of  wire  of  the  same  material  but  of  twice  the  sectional 
area  will  have  a  resistance  of  i  of  3.58  ohms,  or  1.79  ohms. 

From  these  laws  and  definitions,  the  most  common  prob- 
lems and  statements  concerning  electrical  measurements 
will  be  understood. 

EXERCISES.    CXLVIII. 

1.  If  the  resistance  of  700  yds.  of  a  certain  cable  is 
0.91  ohm,  what  is  the  resistance  of  1  mi.  of  that  cable  ? 

2.  The  resistance  of  a  certain  electric  lamp  is  3.8  ohms 
when  a  current  of  10  amperes  is  flowing  through  it.  What 
is  the  voltage  ? 

3.  If  ibhe  resistance  of  130  yds.  of  copper  wire  -^^  in.  in 
diameter  is  1  ohm,  what  is  the  resistance  of  100  yds.  of 
^1^  in.  copper  wire  ? 

4.  The  resistance  of  a  certain  wire  is  9.1  ohms,  and  the 
resistance  of  1  mi.  of  this  wire  is  known  to  be  1.3  ohms. 
Required  the  length. 

5.  Three  arc  lamps  on  a  circuit  have  a  resistance  of 
3.12  ohms  each;  the  resistance  of  the  wires  is  1.1  ohms, 
and  that  of  the  dynamo  is  2.8  ohms.  Find  the  voltage  for 
a  current  of  14.8  amperes. 

6.  The  resistance  of  a  dynamo  being  1.6  ohms,  and  the 
resistance  of  the  rest  of  the  circuit  being  25.4  ohms,  and 
the  electromotive  force  being  206  volts,  find  how  many 
amperes  flow  through  the  circuit. 


RATIO,   VARIATION,    PROPORTION.  323 

Theory  op  Variation. 
If  xccj  and  y  QC  z,  then  xcc  z. 

xccy,  then  x  =  hy.  §  308 

2/  cc  2;,  then  y  =  k'z.  §  308 

X  =  ky  =  kk'z.  Substn. 

xc^z.  §  308 

Note  that  in  step  2  we  cannot  use  the  same  constant  as  in  step  1. 

E.g.,  if  the  edge  of  a  cube  varies  as  the  diagonal  of  a  face,  and  the 
diagonal  of  a  face  varies  as  the  diagonal  of  the  cube,  then  the  edge 
must  vary  as  the  diagonal  of  the  cube. 


321. 

Theorem 

Proof 

1. 

If 

2. 

If 

3. 

.'. 

4. 

322.   Theorem. 

//x 

ocyz,  the?i  y 

ocx/z. 

Proof.    1. 

X  =  kyz. 

(Why  ?) 

2.  .-. 

y  =  \x/z. 

Ax.  7 

3.  .-. 

y  oc  X  /  z. 

§308 

E.g..,  if  the  area  of  a  rectangle  varies  as  the  product  of  the  (numbers 
representing  the)  base  and  altitude,  then  the  base  varies  as  the  quotient 
of  the  (number  representing  the)  area  divided  by  the  (number  repre- 
senting the)  altitude. 

323.    Theorem.  If  w  ocx  and  y  oc  z,  then  wy  oc  xz. 

Proof.    1.  w  =  kx  and  y  =  k'z.                    (Why  ?) 

2.  .-.  wy  =  kk'xz.                             (Why?) 

3.  .".  wyccxz.                                   (Why?) 

E.g.,  it  the  surface  of  a  sphere  varies  as  the  square  of  the  diameter, 
and  ^  of  the  radius  varies  as  the  radius,  then  the  product  of  the  surface 
and  ^  of  the  radius  varies  as  the  product  of  the  radius  and  the  square 
of  the  diameter. 


324  ELEMENTS   OF  ALGEBRA. 

324.  Theorem.  If  xccj  when  z  is  constant,  and  if  x  ccz 
when  J  is  cofistant,  then  x  oc  yz  when  both  j  and  z  vary. 

To  understand  this  statement  consider  a  simple  illustration :  The 
area  of  a  triangle  (p.  172)  varies  as  the  altitude  when  the  base  is  con- 
stant, and  as  the  base  when  the  altitude  is  constant ;  but  it  varies  as 
the  product  of  their  numerical  values  when  both  base  and  altitude  vary. 

Proof.    1.  Let  the  variations  of  y  and  z  take  place  sepa- 
rately. 
2.  Let  X  change  to  x'  when  y  changes  to  y\  z 
remaining  unchanged.     Then 


■  xccy, 


x^^y_ 

x'       y' 


3.  Let  x'  change  to  x"  when  y'  remains  unchanged 
and  z  changes  to  z'.     Then 

x'       z 
x"      z' 

.  X     x'  X  ^      y     z  yz 

4.  .-.  —  •  — >  or  —J  equals  —,-  -?  or  -^^ 

x'    x"  x"  y'    z'  y'z' 

5.  I.e.,  X  changes  to  x"  as  yz  changes  to  y'z',  or 
x  oc  yz. 

Illustrative  problems.  1.  If  xccy,  and  if  x  =  2  when 
y  =  5,  find  x  when  y  =  11. 

•.•  xccy  means  that  x  —  ky,  .-.  2  =  fc  •  5,  and  k  =  f.  .-.  x  =  ly. 
When  ?/  =  11,  X  =  I  •  11  =  4.4. 

2.  The  volumes  of  spheres  vary  as  the  cubes  of  their 
radii.  Two  spheres  of  metal  are  melted  into  a  single 
sphere.      Kequired  its  radius. 

1.  vi  =  kr^  and  v'  =  kr'^.  §  308 

2.  .-.  the  volume  of  the  single  sphere  is  k  (r^  +  r'^). 

3.  Call  v"  this  volume,  and  r"  the  radius  ;   then 

v"  =  k{r^  +  r'^)  =  kr"^. 

4.  .-.  r"^  =  r3  if  r'3,  and  .-.  r"  =  {r^  +  r'^)K 


RATIO,    VARIATION,   PROPORTION.  325 

EXERCISES.    CXLIX. 

1.  li  xocz  and  y  ^z,  prove  that  xy  oc  z"^. 

2.  li  xccz  and  y  ccz,  prove  that  x  +  y  ccz. 

3.  li  x  -{-  y  ccx  —  y,  prove  that  x^  -\-  y^  cc  xy. 

4.  li  w  <x.x  and  y  <^z,  prove  that  w /y  otz  x/z. 

5.  If  10  0?  +  3  ?/  =  7  cc  —  4  2/,  show'  that  xocy. 

6.  If   a^  Gc  h^,    and   if   cc  =  3   when   y  =  5,  prove    that 

7.  If  a?  oc  ?/,  and  if  cc  =  a  when  y  =  b,  find  the  value  of 
X  when  y  =  c. 

8.  If  X  GC  ?/,  and  if  cc  =  7  when  ?/  =  11,  find  the  value 
of  X  when  y  =  7. 

9.  If  xccy,  prove  that  pxcc2)y,  p  being  either  a  con- 
stant or  a  variable. 

10.  What  is  the  radius  of  the  circle  which  is  equal  to 
the  sum  of  two  circles  whose  radii  are  3  and  4,  respectively  ? 

11.  Prove  that  the  volume  of  the  sphere  whose  radius  is 
6  is  equal  to  the  sum  of  the  volumes  of  three  spheres  whose 
radii  are  3,  4,  and  5,  respectively. 

12.  The  illumination  from  a  given  source  of  light  varies 
inversely  as  the  square  of  the  distance.  How  much  farther 
from  an  electric  light  20  ft.  away  must  a  sheet  of  paper  be 
removed  in  order  to  receive  half  as  much  light  ? 

13.  Kepler  showed  that  the  squares  of  the  numbers  rep- 
resenting the  times  of  revolution  of  the  planets  about  the 
sun  vary  as  the  cubes  of  the  numbers  representing  their 
distances  from  the  sun.  Mars  being  1.52369  as  far  as  the 
earth  from  the  sun,  and  the  time  of  revolution  of  the  earth 
being  365.256  das.,  find  the  time  of  revolution  of  Mars. 


826  ELEMENTS   OF  ALGEBRA. 


III.     PROPORTION. 

325.  The  equality  of  two  ratios  forms  a  proportion. 

Thus,  f  =  f,  a  :  b  =  c  :  d,  x/y  =  m/n,  are  examples  of  proportion. 
The  symbol  : :  was  formerly  much  used  for  = . 

326.  There  may  be  an  equality  of  several  ratios,  as 
1:2  =  4:8  =  9:1 8,  the  term  continued  proportion  being 
applied  to  such  an  expression. 

Three  quantities,  a,  b,  c,  are  said  to  be  in  continued  proportion 
when  a  :  b  =  b  :  c. 

327.  There  may  also  be  an  equality  between  the  products 
of  ratios,  as  §  •  f  =  |  •  -y-,  such  an  expression  being  called  a 
compound  proportion. 

328.  In  the  proportion  a  :  b  =  c  :  d,  a,  b,  c,  d  are  called 
the  terms,  a  and  d  being  called  the  extremes  and  b  and  c  the 
means.    The  term  d  is  called  the  fourth  proportional  to  a,  b,  c. 

329.  In  the  proportion  a:b  =  b:c,  b  is  called  the  mean 
proportional  between  a  and  c,  and  c  is  called  the  third  pro- 
portional to  a  and  b. 

330.  If  one  quantity  varies  directly  as  another,  the  two 
are  said  to  be  directly  proportional,  or  simply  proportional. 

E.g.,  at  retail  the  cost  of  a  given  quality  of  sugar  varies  directly  as 
the  weight;  the  cost  is  then  proportional  to  the  weight.  Thus,  at 
4  cts.  a  pound  12  lbs.  cost  48  cts. ,  and  4  cts.  :  48  cts.  =  1  lb.  :  12  lbs. 

331.  If  one  quantity  varies  inversely  as  another,  the  two 
are  said  to  be  inversely  proportional. 

E.g.,  in  general,  the  temperature  being  constant,  the  volume  of  a 
gas  varies  inversely  as  the  pressure,  and  the  volume  is  therefore  said 
to  be  inversely  proportional  to  the  pressure. 


RATIO,    VARIATION,    PROPORTION. 


327 


Illustrative  problems.     1.  What  are  the  mean  proportionals 
between  5  and  125  ? 

1.       .  •5  =  ^. 

X      125 

2.  .-.  625  =  x2. 

3.  .-.  ±  25  =  X,  and  both  results  check. 


2.  What  is  the  fourth  proportional  to  1,  5,  9  ? 

1.  1=1 

5      X 
2.    .-.  a;  =3  5  •  9  =  45,  and  the  result  checks. 


3.  What  number  must  be  added  to  the  numbers  1,  6,  7, 
18  so  that  the  sums  shall  form  a  proportion  ? 

1  +x       7  +x 


1. 


Check.     y5^ 


6  +  X      18  +  X 

18  +  19  X  +  x2  =  42  +  13  X  +  x2. 

x  =  4. 


EXERCISES.    CL. 

1.  State  which  of  the  following,  other  things  being  equal, 
are  directly  and  which  are  inversely  proportional : 

(a)  Volume  of  gas,  pressure. 

(b)  Price  of  bread,  price  of  wheat. 

(c)  Distance  from  fulcrum,  weight. 

(d)  Amount  of  work  done,  number  of  workers. 

2.  Given  1.43  :  x  =  4.01 :  2,  find  the  value  of  x. 

3.  Also  m27  :x  =  x:4:S. 

4.  What  are  the  mean  proportionals  between  1  and  1  ? 

5.  Also  between  1  +  ^  and  2(1  —  i),  where  i  =  v  —  1  ? 

6.  What  is  the  third  proportional  to  1  +  i  and  —  2  ? 


328  ELEMENTS  OF  ALGEBRA. 

332.  The  applications  of  proportion  are  found  chiefly  in 
geometry  and  physics.  Other  methods  are  now  generally 
employed  for  business  problems. 

In  the  two  illustrative  examples  below,  the  first  three 
steps  are  explanatory  of  the  statement  of  the  proportion 
and  may  be  omitted  in  practice.  In  the  first  problem  the 
ratios  are  written  in  the  fractional  form  in  order  that  the 
reasons  involved  may  appear  more  readily. 

Illustrative  problems.  1.  The  time  of  oscillation  of  a 
pendulum  is  proportional  to  the  square  root  of  the  number 
representing  its  length ;  the  length  of  a  1-sec.  pendulum 
being  39.2  in.,  what  is  the  length  of  a  2-sec.  pendulum  ? 

1.  Let  X  =  the  number  of  inches  of  length. 

2.  Then  =  the  ratio  of  the  lengths. 

39.2  ^ 

3.  And         I  =  the  ratio  of  the  corresponding  times  of  oscillations. 

4.  '.•  the  time  is  proportional  to  the  square  root  of  the  number 
representing  the  length, 

.      ^    _2 
V39y2  ~  1 ' 

5.  .-.  -^  =  -,  whence  x  =  39.2  •  4  =  156.8.  Axs.  8,  6 

39.2    r 

6.  •.'  X  =  the  number  of  inches,  .-.  the  pendulum  is  156.8  in.  long. 

2.  A  mass  of  air  fills  10  dm^  under  a  pressure  of  3  kg 
to  1  cm^.  What  is  the  space  occupied  under  a  pressure  of 
5  kg  to  1  cm^,  the  temperature  remaining  constant  ? 

1.  Let         X  =the  number  of  dm^  under  a  pressure  of  5  kg  to  1  cm^. 

2.  Then  x :  10==  the  ratio  of  the  volumes. 

3.  And     5  :  3  =  the  ratio  of  the  corresponding  pressures. 

4.  •.•  the  volume  is  inversely  proportional  to  the  pressure, 

.-.  X  :  10  =  3  :  5. 

5.  .-.  X  =  10  •  3  :  5  =  6.  Ax.  6 

6.  •••  X  =  the  number  of  dm^,  .-.  the  space  is  6  dm^. 


RATIO,   VARIATION,   PROPORTION.  329 


EXERCISES.    CLI. 

1.  How  long  is  a  pendulum  which  oscillates  56  times  a 
linute  ? 

2.  A  cube  of  water  1.8  dm  on  an  edge  weighs  how 
lany  kg? 

3.  If  a  pipe  1.5  cm  in  diameter  fills  a  reservoir  in  3.25 
tins.,  how  long  will  it  take  a  pipe  3  cm  in  diameter  to 

ill  it? 

4.  If  a  projectile  8.1  in.  in  length  weighs  108  lbs.,  what 
the  weight  of  a  similar  projectile  9.37  in.  long  ? 

5.  If  a  metal  sphere  10  in.  in  diameter  weighs  327.5  lbs., 
rhat  is  the  weight  of  a  sphere  of  the  same  substance  14  in. 

Rn  diameter  ? 

6.  Of  two  bottles  of  similar  shape  one  is  twice  as  high 
as  the  other.  The  smaller  holds  0.5  pt.  How  much  does 
the  larger  hold? 

7.  If  a  sphere  whose  surface  is  16  tt  cm^  weighs  5  kg, 
what  is  the  weight  of  a  sphere  of  the  same  substance  whose 
surface  is  32  tt  cm^  ? 

8.  If  the  length  of  a  1-sec.  pendulum  be  considered  as 
1  m,  what  is  the  time  of  oscillation  of  a  pendulum  6.4  m 
long?    62.5m  long? 

9.  A  body  weighs  25  lbs.  5000  mi.  from  the  earth's 
center.  How  much  will  it  weigh  4000  mi.  from  the  center  ? 
(Weight  varies  inversely  as  the  square  of  the  distance  from 
the  earth's  center.) 

10.  The  distance  through  which  a  body  falls  from  a  state 
of  rest  is  proportional  to  the  square  of  the,  number  repre- 
senting the  time  of  fall.  If  a  body  falls  176.5  m  in  6  sees., 
how  far  does  it  fall  in  3.25  sees.  ?    in  1  sec.  ?    in  2  sees.  ? 


330  ELEMENTS  OF  ALGEBRA. 


Theory  of  Proportion. 

333.  Theorem.  In  any  proportion  in  which  the  7iumbers 
are  all  abstract,  the  product  of  the  means  equals  the  product 
of  the  extremes. 

Proof.    1.  If  J  =  ->  then,  by  multiplying  by  bd, 

2.  ad  =  be.  Ax.  6 

334.  Theorem.  If  the  product  of  two  abstract  numbers 
equals  the  product  of  two  others,  either  two  may  be  made 
the  means  and  the  other  two  the  extremes  of  a  proportion. 

Proof.    1.  If  ad  =  be,  then,  by  dividing  by  bd. 

Similarly,  -  =  - ,  etc. 
a      c 

335.  Theorem.     If  Siih  =  c:d,  then  a  :  c  =  b  :  d. 

The  proof  is  left  for  the  student. 

The  old  mathematical  term  for  the  interchange  of  the  means  is 
"alternation."  The  first  proportion  is  "taken  by  alternation"  to 
get  the  second.     The  term,  while  of  little  value,  is  still  used. 

336.  Theorem,    i/*  a :  b  =  c  :  d,  then  b  :  a  =  d  :  c. 

The  proof  is  left  for  the  student. 

The  old  mathematical  term  for  this  change  is  "inversion." 

337.  Theorem.     If  a,:h  =  c-.d,  then  a  +  b:b  =  c  +  d:d. 

The  proof  is  left  for  the  student. 

The  old  mathematical  term  for  this  change  is  "  composition." 

338.  Theorem.    7/"  a :  b  =  c  : d,  then  a  —  b:b  =  c  —  d:d. 

The  proof  is  left  for  the  student. 

The  old  mathematical  term  for  this  change  is  "division." 


RATIO,   VARIATION,   PROPORTION. 


331 


339.    Theorem.     If 

a :  b  =  c  :  d,  then  a.+  b:a  —  b  =  cH-d:c  —  d. 

Proof.    1.  -^  =  -^L_.  §337 

0  d 

2.  ^  =  ^-  §338 

0  d 

_         a-\-b      a  —  h      c-\-d      c  —  d 

4..-.  ^  =  '-^-  §161 

a  —  0       c  —  d 

The  old  mathematical  term  for  this  change  is  "composition  and 
ivision." 
There  is  sometimes  an  advantage  in  applying  this  principle  in  solving 
Jtional  equations.     E.g.,  given  the  equation 

x2-3x  +  1  ~ic2  +  4x-2' 
2  x2  2  x2 


6x  -2       -  8x  +  4 
.-.  X  =  0,  or  f . 

340.    Theorem.     The  mean  proportionals  between  two  num- 
bers are  the  two  square  roots  of  their  product, 
a  _x 
X       b 
x"  =  ab.  §  333,  or  ax.  6 


Proof.    1. 


2.  .-. 

3.  .-. 


X 


±Va6. 


Ax.  9 


Illustrative  problems.     1.  li  a:b  =  c:d,  prove  that 
a-\-b-\-c-{-d:  b-\-d  =  c-\-d:  d. 

1.  This  is  true  if  ad -{- bd  +  cd -{- d^  =^  be  +  bd  +  cd -\-  d^.         §  334 

2.  Or  if  ad  =  be.  Ax.  3 

3.  But  ad  =  6c.  §  333 

4.  .-.  reverse  the  process,  deriving  step  1  from  step  3,  and  the  origi- 
nal proportion  from  step  1. 


332  ELEMENTS  OF  ALGEBRA. 


o    a  1      ^1,            ^-        Va;  +  2  +  Va;  -  3       . . 
2.  Solve  the  equation  -  ,  ,  =  l-*-. 

V^+2  -  Vic  -  3 

We  may  clear  of  fractions  at  once,  isolate  the  two  radicals,  and 
square ;   but  in  this  and  similar  cases  §  339  can  be  used  to  advantage. 
Writing  the  second  member  |  and  applying  §  339,  we  have 


2  Va  +  2  _  5^ 
2  Vx  -  3  ~  1 

2....  ^  =  25. 

X  —  3 

3.    .'.  x  +  2  =25x-75. 

4        .  -7.  _   'TT 

*•      •  •  ^  —   2¥- 

Check.     Substitute  ||  for  x  in  the  original  equation,  and  reduce ; 

then  ^  ^  _  1 

— 7=  —  l^- 

2Vf 

3.  Find  a  mean  proportional  between  1  +  ^  and  —  2  — 14  i. 


1.    By  §  340  this  equals      ±  V(l  +  i)  ( -  2  -  14  i) 


2.  =±Vl2-16i 

3.  =  ±  2  V3-4i 


4.  =  ±  2  V4  -  2  V-  4  -  1 

5.  =  ±  2  (2  -  i).  §  245 

EXERCISES.    CLII. 

1.  Find  the  value  of  ic  in  2  :  3  +  i  =  cc  :  5. 

2.  Find  the  third  proportional  to  1  —  V2  and  1  —  3  V2. 

^4         ^4 

3.  If  a:b  =  c:d,  prove  that  a^ -\-b^\c^-{-d^  =  — -—: ;• 

a-\-o   c-\-d 

4.  Also  that  f^  =  -/ 

b  +d      d 

5.  Also  that  be  -{-  cd:c  —  a  =  *ihcd  +  cd^  —  da^ :  c<^  —  &c. 

6.  Also  that  Va  —  b  :  ^c  —  d  —  V7i  —  Vi  :  Vc  —  V5. 


II 


RATIO,    VARIATION,    PROPORTION.  333 

7.    li  a:h  =  h:c,  prove  that  a  -\-  c>2b. 


8.  li  a:b  =  b:c^  prove  that  (a  -\-  G)b  is  a  mean  propor- 
tional between  a^  +  b'^  and  Z»^  +  c^. 

9.  Find  the  two  mean  proportionals  between 
(a)  2  and  98.  (b)  50  and  -  2. 
(c)  3  and  432.                    (d)  -  7  and  -  847. 

10.    Given  1Q>  —  ^  x  :3  =  2  +  x  :  x,  to  find  x. 


\ 


\ 


11.  Given  Vic  +  7  +  Vcc  —  7  :  Vic  +  7  —  Vx  —  7  =  6:1, 
to  find  cc. 

12.  Given  1  +  x  .13  -  x  =  x  —  2  \ x^ -21  =  x  +  4.:S1  - x^, 
to  find  ic. 

13.  Given  a  — 5:^_    /   -l=a;;6tH-^H 7'  to  find  ic. 

2ab  a  —  b 

14.  Given  3a2  +  2a6-8^>2:5a2  +  4a^'-12^»2  =  x:5a-6^>, 
to  find  x. 

r^-  ^         ab  ^         ab  ^  „ 

15.  Given  X'.y^aArb —  :  a  —  b-\ ->  and  x-\-y:a^ 

=  2  : 1,  to  find  x  and  y. 

16.  Given  Va;  —  5  :  V7  +  x  =  1 :  2,  to  find  x. 

17.  Find  the  value  of  x  in 

^  +  4tx  -  x""  :^  -  4.  X  +  x"^  =  2  +  X  :2  -  X. 

ax  +  cy      ay  -\-  cz      az  +  ex 

18.  It  ; 7^  =  ~ — -—  = — )  prove  that  each  of 

by  -\-  dz      bz  -\-  ax       bx  -{-ay 

these  ratios  equals ;• 

^  b  +  d 

^^    a  —  b  b  —  c         c  —  a  a  -{-b  +  c 

19.  If T~  —  ~^ = = ■, >  prove 

ay  -\-  bx      bz  -\-  ex      ey  -\-  az      ax  -\-  by-j-  ez 

that  each  of  these  ratios  equals 

x  +  y  +  z 


CHAPTER    XVIII. 

SERIES. 

341.  A  series  is  a  succession  of  terms  formed  according 
to  some  common  law. 

E.g.,  in  the  following,  each  term  is  formed  from  the  preceding  as 
indicated : 

1,  3,  5,  7,  ••■,  by  adding  2; 

7,  3,   —  1,   —  5,  •  •  • ,  by  subtracting  4,  or  by  adding  —  4  ; 
3,  9,  27,  81,  •  •  •,  by  multiplying  by  3,  or  by  dividing  by  ^; 

2,  2,  2,  2,  ■  • . ,  by  adding  0,  or  by  multiplying  by  1 . 

In  the  series  0,  1,  1,  2,  3,  5,  8,  13,  •  •  •,  each  term  after  the  first 
two  is  found  by  adding  the  two  preceding  terms. 

342.  An  arithmetic  series  (also  called  an  aritlimetic  pro- 
gression) is  a  series  in  which  each  term  after  the  first  is 
found  by  adding  a  constant  to  the  preceding  term. 

E.g..,  —  7,  —  1,  5,  11,  •  •  •,  the  constant  being  6, 

2,  2,  2,  2,  ...,     "  "  "     0, 

98,  66,  34,  2,  ••  •,     "  "  "      -32. 

343.  A  geometric  series  (also  called  a  geometric  progres- 
sion) is  a  series  in  which  each  term  after  the  first  is  formed 
by  multiplying  the  preceding  term  by  a  constant. 

E.g.,  3,  -  6,  +12,  -  24,  •  •  •,  the  constant  being  -  2, 
10,  5,  2i,  U,  ...,     "  "  "      i, 

2,  2,  2,  2,  ...,     "  "  "      1. 

344.  The  terms  between  the  first  and  last  are  called  the 
means  of  the  series. 

334 


SERIES.  335 

I.     ARITHMETIC   SERIES. 

!345.    Symbols.     The  following  are  in  common  use : 

n,  the  number  of  terms  of  the  series. 

•'    sum         "      "         "  " 

fij  hi  hi  • '  •  tni  the  terms  of  the  series. 

tn  particular,  a,  or  <i,  the '1st  term,  and  Z,  or  ^„,  the  jith  or  last  term, 
d,  the  constant  which  added  to  any  term  gives  the  next ;  d  is  usually 
called  the  difference. 

346.    Formulas.      There  are  two  formulas  in  arithmetic 
series  of  such  importance  as  to  be  designated  as  fundamental. 

1.  t^,  or  I  z=  a  +  (ji  —  1)  d. 

Proof.     1.  ^2  =  a  +  (i,  by  definition. 

tz  =  t2-\-d  =  a-\-2d. 
t^  =  tz-\-d  =  a  +  Zd. 

2.  .-.  tn  =  tn-i-\-d  =  a  +  (n  -  1) d. 

3.  Or  l  =  a  +  {n-\)d. 

E.g.,  the  50th  term  in  the  series  2,  7,  12,  17,  •  •  •  is  2  +  49.5  =  247. 

Z  Li 

Proof.     1.    s  =  a  +  (a  +  (Z)  +  (a  +  2  d)  +  •  •  •  (?  -  d)  +  I. 

2.  Hence,    s^l  -\- {I  -  d)  +  {I  -  2  d)  +  ■  •  •  (a  +  d)  +  a, 
by  reversing  the  order. 

3.  .-.  2  s  =  (a  +  Z)  +  (a  +  0  +  •  •  •  (a  +  0-  Ax.  2 

4.  .-.  2  s  =  n  (a  +  Z),  •••  there  is  an  (a  +  I)  in  step  3  for  each  of  the 
n  terms  in  step  1. 

E.g.,  the  sum  of  the  first  50  terms  of  the  series  2,  7,  12,  17,  •••, 
of  which  I  has  just  been  found,  is 

60(2  +  247)^g^^g 
2 


ELEMENTS  OF  ALGEBRA. 


347.  It  is  evident  that  from  formulas  I  and  II  various 
others  can  be  deduced. 

E.g.,  given  d,  Z,  s,  to  find  n.  The  problem  merely  reduces  to  that 
of  eliminating  a  from  I  and  II,  and  solving  for  n. 

1.  From  I,  a  =  1  —  {n  —  \)d. 

2.  Sabstitutlng  in  11.  s  =  "P'-'^" -'>"]. 

3.  ...  „._^l±i'.„  +  L^  =  0. 

d  d 

2d 

Illustrative  problems.  1.  Which  term  of  the  series  25,  22, 
19,  .  • .  is  -  125  ? 

1.  Given      a  =  25,  d=  -S,  1=  -  125,  to  find  n. 

2.  •.•  l  =  a  +  {n-l)d,   -  125  =  25  +  (n  -  1)  (- 3). 

3.  Solving,  n  =  51! 

2.  Insert  arithmetic  means  between  5  and  41  so  that  the 
4th  of  these  means  shall  have  to  the  next  to  the  last,  less  1, 
the  ratio  1:2. 

1.    The  means  are  5  +  d,  5  +  2  (?,  •  •  •  41  -  2  d,  41  -  d. 
2      .  5  +  4(Z      ^1 

41  -2d  -  1      2' 

3.  .-.  d  =  S,  and  the  means  are  8,  11,  14,  17,  •  •  •  35,  38. 

3.  The  sum  of  three  numbers  of  an  arithmetic  series  is 
12  and  the  sum  of  their  squares  is  56.     Find  the  numbers. 

In  this  and  similar  cases  it  is  advisable  to  take  x  —  y,  x,  x  -\-y,  y 
being  the  common  difference.  In  the  case  of  four  numbers  it  is  advis- 
able to  take  X  —  oy,  X  —  y,  X  +  y,  X  -]-^y,2y  being  the  difference. 

1.  {x-y)^x+{x  +  y)  =  12,  .-.  X  =  4. 

2.  (x  -  vY  +  x2  +  (x  +  2/)2  =  56,  .-.  3  x2  +  2  2/2  =  56. 

3.  .-.  y=±2. 

4.  .-.  the  numbers  are  4  ^  2,  4,  4  ±  2  ;  that  is,  2,  4,  6,  or  6,  4,  2. 


SERIES. 


337 


348.  The  following  table  gives  the  various  formulas  of 
'arithmetic  series,  and  these  should  be  worked  out  from 
formulas  I  and  II  by  the  student. 


Given. 

To  FIND. 

Result. 

1 

2 
3 
4 

a  d  n 
ads 
an  s 
dns 

I 

l  =  a-\-{n-l)d. 

l^-id±V{a-id)^  +  2ds. 

I  =  2s/n-  a. 

l  =  s/n  +  {n-  l)d/2. 

5 
6 

7 
8 

9 
10 
11 
12 

a  d  n 
adl 
a  n  I 
dnl 

S 

s  =  in[2a  +  {n-l)d]. 
s  =  i{l-\-a)  +  {l2-a^)/2d. 
s  =  in{a-^l). 
s  =  in[2l~{n-l)dl 

dnl 
dns 
dls 
n  I  s 

a 

a  =  l-{n-l)d. 

a  =  s/n  —  ^{n  —  l)d. 

a  =  id±V{l  +  idy^-2ds. 

a  =  2 s/n  -I. 

1.3 
14 
15 
16 

a  n  I 
a  n  s 
als 
n  I  s 

d 

d  =  {l-a)/{n-l). 
d  =  2(s-an)/(w2  -  n). 
d  =  {l^  -a^)/{2s-l-a). 
d  =  2{nl  -s)/(n2  -n). 

17 
18 
19 
20 

adl 
ads 
a  I  s 
dls 

n 

n  =  {l  -a-\-d)/d. 

n  ^  [d  -  2  a  ±V{2  a  -  df  +  8  ds]  /2  d. 
n  =  2s/{a  +  l). 

n  =  [d  ^  21  ±V{21  +  d)^  -  Sds]/2d. 

ELEMENTS   OF  ALGEBRA. 


Illustrative  problem.  Find  the  number  of  terms  in  the 
arithmetic  series  whose  first  term  is  25,  difference  —  5,  and 
sum  45. 

We  may  substitute  in  formula  18,  but  it  is  quite  as  easy  to  use  the 
two  fundamental  formulas  which  the  student  will  carry  in  his  mind. 

1.  From  I,  fi=25  +  (n-l)(-5)  =  30-6w. 

2.  "      II,      45  = n. 

2 

3.  .-.  w2  -  11  n  +  18  =  0. 

4.  .-.  {n  -  2)  (n  -  9)  =  0,  and  n  =  2,  or  9. 

The  explanation  of  the  two  results  appears  by  writing 
out  the  series. 

25,  20,  (15,  10,  5,  0,   -  5,   -  10,  -  15). 
The  part  enclosed  in  parentheses  has  0  for  its  sum. 
Hence,  the  sum  of  2  terms  is  the  same  as  the  sum  of  9  terms. 


EXERCISES.    CLIII. 

1.  Find  ^200  ill  the  series  1,  3,  5,  •  •  • . 

2.  Find  s,  given  a  =  4:0,  n  =  101,  d  =  5. 

3.  Find  s,  given  a  =  1,  I  =  200,  n  =  200. 

4.  Given  t^  =  —  li  and  t^^  =  59^,  find  d. 

5.  Find  t^o  in  the  Series  540,  480,  420,  •  •  • . 

•  6.    Find  n,  given  s  =  29,000,  a  =  4.0,  I  =  540. 

7.  Insert  7  arithmetic  means  between  —  5  and  11. 

8.  Insert  12  arithmetic  means  between  —  18  and  125. 

9.  Find  s,  given  a  =  14:,  n  =  S,  d  =  —  4:.     Write  out 
the  series. 

10.    How  many  multiples  of  17  are  there  between  350 
and  1210  ? 


SERIES.  339 

11.  What  is  the  sum  of  the  first  200  numbers  divisible 
by  5  ?  by  7  ? 

12.  Show  that  the  sum  of  any  2n-\-l  consecutive  integers 
is  divisible  by  2  ?i  +  1. 

13.  What  is  the  sum  of  the  first  50  odd  numbers  ?  the 
first  100  ?  the  first  n  ? 

14.  What  is  the  sum  of  the  first  50  even  numbers  ?  the 
first  100  ?  the  first  n  ? 

15.  Given  1  =  11,  d  =  2,  s  =  32,  to  find  7i.  Check  the 
result  by  writing  out  the  series. 

16.  How  long  has  a  body  been  falling  when  it  passes 
through  53.9  m  during  the  last  second  ? 

17.  Suppose  every  term  of  an  arithmetic  series  to  be 
multiplied  by  A; ;    is  the  result  an  arithmetic  series  ? 

18.  The  sum  of  four  numbers  of  an  arithmetic  series  is 
0  and  the  sum  of  their  squares  is  20.     Find  the  numbers. 

19.  The  sum  of  four  numbers  of  an  arithmetic  series  is 
12  and  the  sum  of  their  squares  is  116.     Find  the  numbers. 

20.  The  sum  of  three  numbers  of  an  arithmetic  series 
is  21  and  the  sum  of  their  squares  is  179.  Find  the 
numbers. 

21.  Find  five  numbers  of  an  arithmetic  series  such  that 
the  sum  of  the  first  and  fifth  is  46,  and  that  the  ratio  of  the 
fourth  to  the  second  is  1.3. 

22.  $100  is  placed  at  interest  annually  on  the  first  of 
each  January  for  10  yrs.,  at  6%.  Find  the  total  amount  of 
principals  and  interest  at  the  end  of  10  yrs. 

23.  Find  the  7ith.  term  and  the  sum  of  the  first  n  terms  : 
(a)  1  +  34.^  +  ....  (b)   11  +  9  +  7+  .... 


340"  ELEMENTS   OF  ALGEBRA. 


II.     GEOMETRIC   SERIES. 

349.  Symbols.     The  following  are  in  common  use : 

w,  s,  a,  I  and  ti,  t^,  ■  •  •  tn,  as  in  arithmetic  series  ; 
r,  the  constant  by  which  any  term  may  be  multiplied  to  produce 
the  next ;  r  is  usually  called  the  rate  or  ratio. 

350.  Formulas.  There  are  two  formulas  in  geometric 
series  of  such  importance  as  to  be  designated  as  fundar 
mental. 

I.    t^,  01  I  =  ar^'-K 

Proof.    1.  ^2  =  ctr,  by  definition. 

^3  =  t2r       =  ar^. 
ti  =  t^r       =  ar^. 


2.  .-.  tn  =  tn-ir  —  ar''-'^. 

3.  Or  1  =  ar»-i. 

S.g.i  the  7th  term  of  the  series  16,  8,  4,  •  •  •  is 


_  ^^"  —  a  _  Ir  —  a 


Proof.     1.  s  =  a  -^  ar  -\-  ar^  +  •  •  •  +  ar^—^  +  ar"^~^. 

2.  .-.  rs  —         ar  -\-  ar^  +  •  •  •  +  ar"^—^  +  ar'^  —  ^  +  ar^^ 
by  multiplying  by.r. 

3.  .-.      rs  —  s  =  ar^  —  a,  by  subtracting,  (2)  —  (1). 

dyn  Q^ 

4.  .-.  {r  —  1)  s  =  ar^  —  a,  and  s  = ,  by  dividing  by  (r  —  1). 

p      *    J  1  7  Ir  —  a 

5.  And  •••  ar^  =  ar'^-^  •  r  =  Ir,  .-.  s  = 

r-1 

E.g.,  the  sum  of  the  first  7  terms  of  the  series  16,  8,  4,  •  •  • ,  of  which 
I  has  just  been  found,  is 


SERIES. 


341 


351.    It  is  evident  that  from  formulas  I  and  II  various 
)thers  can  be  deduced. 

1 
E.g.^  giveu  Z,  a,  n,  to  find  r.     •.•  I  =  ar^—\  .-,  r  =  {l/a)»—\ 

Given  n,  Z,  s,  to  find  a.     The  problem  reduces  to  that  of  eliminating 
from  I  and  II  and  solving,  if  possible,  for  a. 

1.  From  II,  r  = 

2.  Substitute  this  in 


I,  and       z  =  a(i-|y    \ 
a{s  —  a)"-! 


0. 


Here  it  is  impossible  to  isolate  a.     When  the  numerical  values  of 
s,  w  are  given,  a  can  frequently  be  determined  by  inspection. 

Eor  example,  given  w  =  4,  Z  =  8,  s  =  15,  to  find  a.     Here 
.8-73  =  a(15-a)3, 

and  a  evidently  equals  8,  or  1.     Either  value  checks,  for  the  series 
may  be  8,  4,  2,  1,  or  1,  2,  4,  8. 


Illustrative  problems.  1.  Find  the  sum  of  five  consecutive 
powers  of  3,  beginning  with  the  first. 

1.  Here  a  =  S,  r  =  S,  n  =  6. 

2.  s  =  (a>-«  -  a)  /  (r  -  1)  =  (3  •  35  -  3)  /2  =  363. 

2.  Of  three  numbers  of  a  geometric  series,  the  sum  of 
•the  first  and  second  exceeds  the  third  by  3,  and  the  sum 
of  the  first  and  third  exceeds  the  second  by  21.  Find  the 
numbers. 

1.  Let  X,  xy,  xy^  be  the  numbers. 

2.  Then  x  +  xy  =  xy^  +  3,  or  x  +  xy  —  S  =  xy^. 

3.  And  X  +  xy^  =  xy  +  21,  or  —  x  -{■  xy  +  21  =  xy^. 

4.  .-.  x  +  xy  —  S  =  —  X  -{■  xy  +  21,  or  x  =  12. 

5.  .-.  4  ?/2  _  4  y  _  3  =  0,  by  substituting  in  2. 

6.  .-.  {2y  +  l){2y-S)  =  0,  and  y=-  1,  or  f. 

7.  .-.  the  numbers  are  12,  —  6,  3,  or  12,  18,  27.     Each  set  checks. 


342 


ELEMENTS  OF  ALGEBRA. 


352.  The  following  table  gives  the  various  formulas  of 
geometric  series.  They  should  be  worked  out  from  formulas 
I  and  II  by  the  student,  excepting  those  for  n.  The  for- 
mulas for  n  require  logarithms  and  may  be  taken  after 
Chap.  XIX. 


GiVKN. 

To  FIND. 

Results. 

1 

ar  n 

I  =  ar''-\ 

2 

ar  s 

I 

l=[a  +  {r-l)s]/r. 

3 

a  71  s 

Z(s-0«-' -a(s-a)"-i=0. 

4 

r  n  s 

l  =  {r-l)srn-^/{rn-l). 

5 

ar  n 

s  =  a(r«-l)/(r-l). 

6 

7 

an 

a  n  I 

s 

s  =  (W-a)/(r-l). 

71                           «                           1                             1 

8 

r  n  I 

s  =  Z(r»-l)/(r«-r»-i). 

9 

r  n  I 

a  =  l/r»-K 

10 

r  n  s 

a  =  8(r-l)/(r--l). 

11 

rls 

a 

a  =  rl-{r  -  l)s. 

12 

n  I  s 

Z  (s  _/)«-!_  a  (s-a)«-i  =  0. 

13 

a  n  I 

r=  {l/a)^>^^. 

14 

an  s 

r^-sr/a  +  {s-a)/a  =  0. 

15 

a  I  s 

r  =  {s-a)/{s-l). 

16 

n  I  s 

rn  _  sr^-'^/{s  -l)  +  l/{s-l)  =  0. 

17 

arl 

n  =  (log  I  -  log  a)  /log  r  +  1. 

18 

ar  s 

n  =  {log[a  +  (r- l)s] -loga}/logr. 

19 

a  I  s 

n 

n  =  (log  I  -  loga)/[log  (s-a)-  log  (s  - 1)]  +  1. 

20 

rls 

n  =  {log  Z  -  log  [ir  -  (r  -  1)  s] }  /log  r  +  1 . 

SERIES.  343 

EXERCISES.    CLIV. 

1.  The  sum  of  how  many  terms  of  the  series  4,  12, 
S6,    -is  118,096? 

2.  Find  the  sum  of  the  first  ten  terms  of  the  series 
3^, -2^|.3V-- 

3.  Find  the  geometric  mean  between 
(a)  1  and  4.  (b)   -  2  and  -  8. 

4.  Find  the  sum  of  five  numbers  of  a  geometric  series, 
the  second  term  being  5  and  the  fifth  625. 

5.  What  is  the  fourth  term   of   the  geometric  series 
I  whose  first  term  is  1  and  third  term  ^V  ? 

6.  The  arithmetic  mean  between  two  numbers  is  39  and 
the  geometric  mean  15.     Find  the  numbers. 

7.  Prove  that  the  geometric  mean  between  two  numbers 
is  the  square  root  of  their  product  (§  343). 

8.  Prove  that  the  arithmetic  mean  between  two  unequal 
positive  numbers  is  greater  than  the  geometric  mean. 

9.  To   what    sum   will    $1   amount   at  4^    compound 
interest  in  5  yrs.  ?     (Here  a  =  $1,  r  =  1.04,  n  =  6.) 

10.  In  ex.  9,  suppose  the  rate  were  4%  a  year,  but  the 
interest  compounded  semiannually  ? 

11.  The  sum  of  the  first  eight  terms  of  a  certain  geo- 
metric series  is  17  times  the  sum  of  the  first  four  terms. 
What  is  the  rate  ? 

12.  Find  the  10th  term  and  the  sum  of  the  first  ten 
terms  of  the  series: 

(a)  l,hh---  (b)  1,-2,4,-8,.... 

(c)  1,2,4,....  (d)  32,-16,8,-4,.... 


344  ELEMENTS   OF   ALGEBRA. 

353.    Infinite  geometric  series.     If  the  number  of  terms  is 

infinite  and  r<l,  then  s  approaches  as  its   limit 

(§  167).  ^  ~  '' 

This  is  indicated  by  the  symbols  s  = ,  n  being  infinite. 

1  —  r 

The  symbol  ==  is  read  "  approaches  as  its  limit "  (p.  140). 

Proof.     1.  ■.*  7'  <  1,  the  terms  are  becoming  smaller,  each 
being  multiplied  by  a  fraction  to  obtain  the  next. 

2.  .'.1  =  0,  and  .*.  Ir  =  0,  although  they  never  reach  that 
limit. 

3.  .'.  s  == }  by  formula  II. 

r  —  1       ^ 

4.  .-.8  =  - J  by  multiplying  each  term  of  the  frac- 
tion by  —  1. 

E.g.,  consider  the  series  1,  i,  i,  •••,  where  n  is  infinite.     Here 
a  1 

5  =: ,  or  - — —  ,  or  2.     That  is,  the  greater  the  number  of  terms, 

1  —  r  1  —  i 

the  nearer  the  sum  approaches  2,  although  it  never  reaches  it  for 
finite  values  of  n. 


EXERCISES.    CLV. 

1.  Given  s  =  8,  a  =  4:.     Find  r. 

2.  Given  s  =  10^,  r  =  ^.     Find  a. 

3.  Given  s  =  l,r  =  §§99.     Find  a. 

4.  Given  s  =  155,  r  =  2,  n  =  5.     Find  a. 

5.  Given  s  =  124.4,  r  =  S,  n  =  4^.     Find  a. 

6.    Find  the  limits  of  the  following  sums,  n  being  infinite : 
(a)  20  +  10  +  5  +  2^  +  ....      (b)  |  +  i  +  /,,4-T^^  +  ---. 

(d)  10  +  1  +  0.1  +0.01  +  .... 


SERIES.  346 

354.    Circulating  decimals.     If  the  fraction  y\  is  reduced 

the  decimal  form,  the  result  is  0.272727-  •  • ,  and  similarly 
le   fraction   |^  =  0.152777- •  •.      The   former   constantly 
jpeats  27,  and  the  latter  constantly  repeats  7  after  0.152. 
When,  beginning  with  a  certain  order  of  a  decimal  frac- 
jtion,  the  figures  constantly  repeat  in  the  same  order,  the 
lumber  is  called  a  circulating  decimal,  and  the  part  which 
3peats  is  called  a  circulate. 
A  circulate  is  represented  by  a  dot  over  its  first  and  last  figures. 
0.272727  •  -  •  is  represented  by  0.27  ; 
0.152777---  "  "  "    0.1527. 

A  circulating  decimal  may  be  reduced  to  a  common  frac- 

ion  by  means  of  the  formula  s  =  - >  as  follows  : 

•^  1  —  r 

1.  To  what  common  fraction  is  0.27  equal  ? 

1.  0.27  =  0.27  +  0.0027  +  0.000027  +  -  •  •  • 

2.  This  is  a  geometric  series  with  a  =  0.27,  r  =  0.01,  n  infinite. 
0.27  27       3 


3. 


1  -  0.01      99      11 


2.  To  what  common  fraction  is  0.1527  equal 


1.   0.1527  =  0.152  +  0.0007  +  0.00007  +  -  -  •  =  0.152  +  a  geometric 
series  with  a  =  0.0007,  r  =  0.1,  n  infinite. 
^  0.0007  7 

'    "^"1  -0.1~9000' 
3.   To  this  must  be  added  0.152,  giving  0.152|,  or  if^§,  or  ^. 

EXERCISES.    CLVI. 

Express  as  common  fractions : 

1.    0.3.  2.    0.045.  3.    O.OOOl. 

4.   0.147.  5.    1.2375.  6.   5.0504. 

7.    0.045.  8.   2.003471.  9.    0.23456. 


346  ELEMENTS   OF  ALGEBRA. 


III.     MISCELLANEOUS   TYPES. 

355.  Of  the  other  types  of  series,  some  can  be  treated  by 
the  methods  which  have  just  been  considered. 

Illustrative  problems.  1.  Defining  a  harmonic  series  as  one 
the  reciprocals  of  whose  terms  form  an  arithmetic  series, 
insert  three  harmonic  means  between  2  and  4. 

This  reduces  to  the  insertion  of  three  arithmetic  means  between  | 
and  ^. 

1.  •••  a  =  i,  n  =  6,  and  I  =  i, 

2.  .-.  |,  =  ^  +  4d,  and  d=-j\. 

3.  .-.  the  arithmetic  series  is  ^,  ^^,  f ,  y\,  J, 
and  "   harmonic         "        2,  2f,  2f,  3^,  4. 

2.  Sum  to  20  terms  the  series  1,  —  3,  5,  —  7,  9,  —  11, ... . 

Here  the  odd  numbers  of  the  terms  form  an  arithmetic  series  with 
d  =  4,  and  the  even  ones  form  an  arithmetic  series  with  d  =  —  4. 
There  are  ten  terms  in  each  set.     Summing  separately,  we  have 

190  -  210  =  -  20. 

3.  What  is  the  harmonic  mean  between  a  and  b  ? 

1.    If  yt  is  the  harmonic  mean,  -,  -,  -  must  form  an  arithmetic 
.      ,       1,  '  a'  h'   b 

series  (ex.  1). 

2..-.  1_1  =  1_1. 

h      a      b      h 

3..:  h^^"" 


a  +  b 


E.g.,  the  harmonic  mean  between  3  and  4  is  Y-  For,  taking  the 
reciprocals  of  3,  -y,  and  4,  we  have  i,  ^j,  |,  or  gj,  -i^,  and  2?,  which 
form  an  arithmetic  series. 

4.  rind  the  sum  of  71  terms  of  the  series  1,  2x,  3x^, 
Ax\  .... 


SERIES.  347 

Here  the  coefficients  form  an  arithmetic  series  and  the  x's  a  geo- 
letric.     Such  a  series  is  oalled  arithmetico-geometric. 
Let  s  =  1  +  2  X  +  3  x2  +  . . .  +  (71  -  1)  x"-2  +  nx«-i ; 

[then  xs=  x  +  2x2  H \- {n  -  2)x»-2  +  (n  -  l)x"-i  +  nx«. 

Subtracting, 

(1  -  x)  s  =  1  +    X  +    x2  +  •  ■  •  +  x«-2  +  x«-i  —  nx". 

_    1  —  X"  x» 

•■■'"(1-X)2~''(1-X)" 

EXERCISES.    CLVII. 

1.  Sum  the  series  3,  6,  •  •  •  3  (?^  —  1),  3  ti. 

2.  Sum  to  2  7i,  terms  the  series  1,  —  2,  +  3,  —  4,  •  •  • . 

3.  Sum  the  series  1,  4 ic,  7 x^,  10  x^,  •  •  -jio  n  terms. 

4.  Sum  the  series  1,  —  3,  +5,  —  7,  H to  2n  terms. 

5.  Insert  a  harmonic  mean  between  2  and  2 ;  between 
-2  and  -2. 

6.  Prove  that  no  two  unequal  numbers  can  have  their 
arithmetic,  geometric,  and  harmonic  means  equal,  or  any 
two  of  these  equal. 

7.  Show  that  the  sum  of  the  first  n  terms  of  the  series 
1,  —  2,  +4,  —  8,  H-  16,  •  •  •  is  ^  (1  ±  2"),  the  sign  depending 
on  whether  n  is  odd  or  even. 

8.  Find  the  sum  of  1  +  2  a;  +  3  ic^  +  4  cc^  H to  n  terms 

by  writing  the  series  (1  +  a;  +  £c^  H )  -\- (x -\- x^ -\- x^ -] ) 

■}- (x^ -\- x^ -] )  +  (x^  -\ ),  etc.,  summing  each  group  sepa- 
rately, and  adding  the  sums. 

9.  The  number  of  balls  in  a  triangular  pile  is  evidently 

lH-(l  +  2)  +  (l+2  +  3)H ,    depending  on  the  number 

of  layers.     How  many  balls  in  such  a  pile  of  10  layers  ? 


CHAPTER   XIX. 

LOGARITHMS. 

356.  About  the  year  1614  a  Scotchman,  John  Napier, 
invented  a  scheme  by  which  multiplication  can  be  per- 
formed by  addition,  division  by  subtraction,  involution  by 
a  single  multiplication,  and  evolution  by  a  single  division. 

357.  In  considering  the  annexed  series  of  numbers  it  is 
apparent  that 

1.  V  23.25  =  28, 

8  .  32  =  28  =  256. 
.-.  the  product  can  be  found  by  adding  the 
exponents  (3  +  5  =  8)  and  then  finding  what 
28  equals. 

2.  •.•  29  :  23  =  26, 

512  :  8  =  64. 
.-.  this  quotient  can  be  found  from  the  table  by  a  single  subtraction 
of  exponents. 

3.  •.•  (25)2  ^  25  .  25  =  21^ 

322  ^  1024. 

4.  •.•  V2io  =  V25  .  25  =  25, 

Vi024  =  32. 

5.  The  exponents  of  2  form  an  arithmetic  series,  while  the  powers 
form  a  geometric  series. 

In  like  manner  a  table  of  the  powers  of  any  number  may 
be  made  and  the  four  operations,  multiplication,  division, 
involution,  evolution,  reduced  to  the  operations  of  addition, 
subtraction,  multiplication,  and  division  of  exponents. 

348 


20  =  1 

26 

=  64 

21  =  2 

2" 

=  128 

22  =  4 

28 

=  256 

23  =  8 

29 

=  512 

24  =  16 

210 

=  1024 

25  =  32 

211 

=  2048 

I 


LOGARITHMS.  349 


I 


358.  For  practical  purposes^  the  exponents  of  the  powers 
to  which  10,  the  base  of  our  system  of  counting,  must  be 
raised  to  produce  various  numbers  are  put  in  a  table,  and 
these  exponents  are  called  the  logarithms  of  those  numbers. 

In  this  connection  the  word  power  is  used  in  its 
broadest  sense,  10'*  being  considered  as  a  power,  whether  n 
is  positive,  negative,  integral,  or  fractional.  The  logarithm 
of  100  is  written  "  log  100." 

E.g.,  103   :.^1000,  /.  log  1000=3.  102   ^iqo,  .-.  log  100  =2. 

10'>   =1,        .-.  log  1       =0.  101   =  10,  .-.log    10=1. 

10-i=i,      .-.logO.l    =-1.       10-2=  — ,  .-.log  0.01= -2. 
10'  ^  102'  ^ 

\{y?s^6^  that  is,  the  thousandth  root  of  lO^'^i,  is  nearly  2, 
.-.  log  2  =  0.301,  nearly. 

Although  log  2  cannot  be  expressed  exactly  as  a  decimal 
fraction,  it  can  be  found  to  any  required  degree  of  accuracy. 


EXERCISES.    CLVIII. 

1.  What  is  the  logarithm  of  10" « ?    of  1000^  ?    of  10^  ? 

2.  Whatis  the  logarithm  of  10^- 10«?   of  lOMO^  ? 

3.  What  is  the  logarithm  of  -^lO"*  •  10^  ■  10«  ?   of  vio  ? 

4.  What  is  the  logarithm  of  lO^-lO^-lO^?    of  0.001  of 
10-10*?    of  10^   10^.  109? 

5.  Between  what  two  consecutive  integers  does  log  800 
lie,  and  why  ?   also  log  3578  ?   log  27  ? 

6.  Between  what  two  consecutive  negative  integers  does 
log  0.02  lie,  and  why  ?   also  log  0.009  ?   log  0.0008  ? 

7.  If  the  logarithm  of  2  is  0.301,  what  is  the  logarithm 

of  21000  9     (2  =  10^^^%    .•.2"o«  =  ?     .-.   the   logarithm   of 
21000  ^  9\ 


350  ELEMENTS   OF  ALGEBRA. 

359.  Since  2473  lies  between  1000  and  10,000,  its  loga- 
rithm lies  between  3  and  4.  It  has  been  computed  to  be 
3.3932.  The  integral  part  3  is  called  the  characteristic  of 
the  logarithm,  and  the  fractional  part  0.3932  the  mantissa. 

That  is,  lo^BB^^,  or  103-3932     =2473,      .-.log     2473  =  3.3932. 

...  103.3991!.  101=102.8932^  .-.  102-8932     =247.3,     ..  log    247.3  =  2.3932. 

Similarly,  ioi.3982     =24.73,     .-.log   24.73  =  1.3932. 

*"      '"^  100-3932     =2.473,    .-.log   2.473=0.3932. 

100-3932-1=0.2473,  .-.log 0.2473=0.3932-1. 

360.  It  is  thus  seen  that 

1.  The  characteristic  can  always  he  found  hy  inspection. 

Thus,  because  438  lies  between  100  and  1000,  hence  log  438  lies 
between  2  and  3,  and  log  438  =  2  +  some  mantissa. 

Similarly,  0.0073  lies  between  0.001  and  0.01,  hence  log  0.0073  lies 
between  —  3  and  —  2,  and  log  0.0073  =  —  3  +  some  mantissa. 

Since  5  lies  between  1  and  10,  log  5  lies  between  0  and  1,  and  equals 
0  +  some  mantissa. 

2.  The  mantissa  is  the  same  for  any  given  succession  of 
digits,  wherever  the  decimal  point  may  he. 

Thus,  log  2473  =  3.3932,  and  log  0.2473  =  0.3932  -  1. 

3.  Therefore,  only  the  mantissas  need  he  put  in  a  table. 

Instead  of  writing  the  negative  characteristic  after  the  mantissa, 
it  is  often  written  before  it,  but  with  a  minus  sign  above ;  thus,  log 
0.2473  =  0.3932  -  1  =  1.3932,  this  meaning  that  only  the  character- 
istic is  negative,  the  mantissa  remaining  positive. 

Negative  numbers  are  not  considered  as  having  loga- 
rithms, but  operations  involving  negative  numbers  are 
easily  performed,  ^^.g-,  the  multiplication  expressed  by 
1.478  •  (—  0.007283)  is  performed  as  if  the  numbers  were 
positive,  and  the  proper  sign  is  prefixed. 


LOGARITHMS.  351 

EXERCISES.    CLIJC. 

1.  What  is  the  characteristic  of  the  logarithm  of  a 
number  of  three  integral  places  ?    of  6  ?    of  20  ?    of  7i? 

2.  What  is  the  characteristic  of  the  logarithm  of  0.3  ? 
of  any  decimal  fraction  whose  first  significant  figure  is  in 
the  first  decimal  place  ?  the  second  decimal  place  ?  the 
20th  ?    the  nth  ? 

3.  From  exs.  1,  2  formulate  a  rule  for  determining  the 
characteristic  of   the  logarithm  of   any  positive  number. 

4.  If  log  39,703  =-4.5988,  what  are  the  logarithms  of 
"(a)  39,703,000?  (b)  397.03?         (c)  3.9703? 
(d)  0.00039703?         (e)  0.39703?       (f)  3970.3? 

361.    The  fundamental  theorems  of  logarithms. 

I.  The  logarithm  of  the  product  of  two  numbers  equals 
the  sum  of  their  logarithms. 

1.  Let  a  —  10"*,      then  log    a  —  m. 

2.  Let  b  =  10%         "     log    b  =  n. 

3.  .*.    ab  =  10"'  +  ",  and  log  ab  =  m  -{- n  =  log  a  -{-  log  b. 

Thus,  log  (5x6)=  log  5  +  log  6. 

II.  The  logarithm  of  the  quotient  of  two  numbers  equals 
the  logarithm  of  the  dividend  minus  the  logarithm  of  the 
divisor. 

1.  Let  a  =  10"*,  then  log  a  =m. 

2.  Let  b  =  10«,  "     log  b  =  n. 

a       10"'      ^^  ^     .       a 

3.  .-.     -  =  — —  =  lO"*-"  and    log-  =  m  —  n. 

b       10"  0 

Thus,  log  (40  ^  5)  =  log  40  -  log  5. 


352  ELEMENTS  OF  ALGEBRA. 

III.    The  logarithin  of  the  nth  power  of  a  number  equals 
n  times  the  logarithm  of  the  number. 

1.  Let  a  =  lO"*,         then  log   a  =  m. 

2.  .'.    a"  =  lO""*,        and   log  a"  =  nm  =  n  log  a. 


IV.    The  logarithm  of  the  nth  root  of  a  number  equals 
—  th  of  the  logarithm  of  the  number. 

1.  Let  a  =  10"*,         then  log    a  =  m. 

-  -  -ml 

2.  .'.    a"  =  10",         and    log  a"  =  —  =  -  •  log  a. 

n       n 

Th,  III  might  have  been  stated  more  generally,  so  as  to  include 

X 

Th.  IV,  thus :    log  a^  =  -  •  log  a.     The  proof  would  be  substantially 
the  same  as  in  ths.  Ill  and  IV. 


EXERCISES.    CLX. 

Given  log  2  =  0.3010,   log  3  =  0.4771,   log  5  =  0.6990, 
log  7  =  0.8451,  and  log  514  =  2.7110,  find  the  following : 


1. 

log  60. 

2.    log  24. 

3. 

log  7«. 

4. 

log^. 

5.    log  625. 

6. 

log  7*. 

7. 

log  ^3^. 

8.    log-^. 

9. 

log  35. 

10. 

log  5141 

11.    log  1.05. 

12. 

log  257. 

13. 

log  1050. 

14.   log  154,200. 

15. 

log  V'514. 

16. 

log  10.28. 

17.   log  154.2. 

18. 

log  3.598. 

19. 

log  0.3084. 

20.    log  30.84. 

21. 

log  15.421 

22.   log 

1799  [=  log  ft.  514- 

m 

23.    Show  how  to  find  log  5, 

given  log  2. 

LOGARITHMS. 


353 


362.  Explanation  of  table.  Given  a  number  to  find  its 
logarithm.  In  the  table  on  pp.  354  and  355  only  the  man- 
tissas are  given.  For  example,  in  the  row  opposite  71,  and 
under  0,  1,  2,  •  •  •  will  be  found : 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

71 

8513 

8619 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

This  means  that  the  mantissa  of  log  710  is  0.8513,  of 
log  711  it  is  0.8519,  and  so  on  to  log  719.     Hence, 
log       715  =  2.8543,  log      7.18  =  0.8561, 

log  71,600  =  4.8549,  log  0.0719  =  2.8567. 

And  •••  7154  is  ^^  of  the  way  from  7150  to  7160,  .-.log 
7154  is  about  ^-^  of  the  way  from  log  7150  to  log  7160. 
.'.  log  7154  =  log  7150  +  y\  of  the  difference  between 
log  7150  and  log  7160 
=  3.8543  +  T-%  of  0.0006 
=  3.8543  +  0.0002  =  3.8545. 
Similarly,  log      7.154  =  0.8545, 

and  log  0.07154  =  2.8545. 

The  above  process  of  finding  the  logarithm  of  a  number  of 
four  significant  figures  is  called  interpolation.  It  is  merely 
an  approximation  available  within  small  limits,  since  num- 
bers do  not  vary  as  their  logarithms,  the  numbers  forming 
a  geometric  series  while  the  logarithms  form  an  arith- 
metic series.  It  should  be  mentioned  again  that  the  man- 
tissas given  in  the  table  are  only  approximate,  being  cor- 
rect to  0.0001.  This  is  far  enough  to  give  a  result  which 
is  correct  to  three  figures  in  general,  and  usually  to  four, 
an  approximation  sufficiently  exact  for  many  practical  com- 
putations. 


354 


ELEMENTS   OF   ALGEBRA. 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0 

0000 

0000 

3010 

4771 

6021 

6990 

7782 

8451 

9031 

9542 

1 

0000 

0414 

0792 

1139 

1461 

1761 

2041 

2304 

2553 

2788 

2 

3010 

3222 

3424 

3617 

3802 

3979 

4150 

4314 

4472 

4624 

3 

4771 

4914 

5051 

5185 

5315 

5441 

5563 

5682 

5798 

5911 

4 

6021 

6128 

6232 

6335 

6435 

6532 

6628 

6721 

6812 

6902 

5 

6990 

7076 

7160 

7243 

7324 

7401 

7482 

7559 

7634 

7709 

6 

7782 

7853 

7924 

7993 

8062 

8129 

8195 

8261 

8325 

8388 

7 

8451 

8513 

8573 

8633 

8692 

8751 

8808 

8865 

8921 

8976 

8 

9031 

9085 

9138 

9191 

9243 

9294 

9345 

9395 

9445 

9494 

9 

9542 

9590 

9638 

9685 

9731 

9777 

9823 

9868 

9912 

9956 

10 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

11 

0414 

0153 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

12 

0792' 

10828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1106 

13 

1139 

1173 

1206 

1239 

1271 

1303 

1335 

1367 

1399 

1430 

14 

1461 

1492 

1523 

1553 

1584 

1614 

1614 

1673 

1703^ 

1732 

15 

1761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

IG 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

17 

2304 

2330 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

18 

2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

19 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

3222 

3243 

3263 

3284 

3301 

3324 

3345 

3365 

3385 

3404 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

;i5fi0 

3579 

3598 

23 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

26 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

27 

4314 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

29 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

5011 

5024 

5038 

32 

5051 

5065 

5079 

5092 

5105 

5119 

5132 

5145 

5159 

5172 

33 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

34 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

35 

5441 

5453 

5465 

5478 

5490 

5502 

5514 

5527 

5539 

■5551 

36 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5&58 

5670 

37 

5682 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

5775 

5786 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

39 

5911 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

42 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

0425 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

45 

6532 

6542 

6551 

6561 

6571 

6580 

&590 

6599 

6609 

6618 

46 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

49 

6902 

6911 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

LOGARITHMS. 


355 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7143 

7152 

52 

7160 

71 G8 

7177 

7185 

7193 

7202 

7210 

7218 

7226 

7235 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

7316 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

7380 

7388 

7396 

55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

56 

7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

57 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

58 

7634 

7642 

7019 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

7980 

7987 

63 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

64 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116 

8122 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

68 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627 

73 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8818 

8854 

8859 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9133 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

9186 

83 

9191 

9196 

9201 

9206 

9212 

9217 

922"^ 

9227 

9232 

9238 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

86 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

88 

9445 

9450 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

89 

9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

91 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

92 

9638 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 

93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

94 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

96 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

97 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

98 

9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 

N 

0 

1 

2 

3 

4 

5 

G 

7 

8 

9 

356 


ELEMENTS   OF  ALGEBRA. 


In  all  work  with  logarithms  the  characteristic  should  he 
writte?i  before  the  table  is  consulted,  even  if  it  is  0.  Other- 
wise it  is  liable  to  be  forgotten,  in  which  case  the  computa- 
tion will  be  valueless. 

Illustrative  problems.     1.  Find  from  the  table  log  4260. 
The  characteristic  is  3. 

The  mantissa  is  found  to  the  right  of  42  and  under  6  ;  it  is  0.6294. 
.-.  log  4260  =  3.6294. 

2.  Find  from  the  table  log  42.67. 
The  characteristic  is  1. 

log  42.7  =  1.6304 

log  42.6  =  1.6294 

difference  =  0.0010 

tV  of  0.0010  =  0.0007 

.-.  log  42.67  =  1.6294  +  0.0007 

=  1.6301. 


EXERCISES.    CLXI. 

From  the  table  find  the  following  : 


1.  log  28. 

4.  log  2.34. 

7.  log  8940. 

10.  log  3855. 

13.  log  1003. 

16.  log  23.42. 

19.  log  75.551 

22.  log  0.2969. 


2.  log  443. 

5.  log  6.81. 

8.  log  43.41. 

11.  log  2.005. 

14.  log  3.142. 

17.  log  Vl28. 

20.  log  0.0007. 

23.  logO.01293. 


3.  log  9.823. 

6.  log  700.3. 

9.  log  V^125. 

12.  log  9.8211 

15.  log  24,000. 

18.  log  0.2346. 

21.  log  0.00323. 

24.  log  0.000082. 


LOGARITHMS. 


357 


363.  Given  a  logarithm,  to  find  the  corresponding  number. 
The  number  to  which  a  logarithm  corresponds  is  called  its 
antilogarithm. 

E.g.,  :•  log 2  =  0.3010,     .-.  antilog 0.3010  =  2. 

The  method  of  finding  antilogarithms  will  be  seen  from 
a  few  illustrations.  Referring  again  to  the  row  after  71 
on  p.  355,  we  have : 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

71 

8613 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

Hence,  we  see  that 
antilog  0.8513  =  7.1,  antilog  5.8531  =  713,000, 

antilog  2.8567  =  0.0719,  antilog  1.8555  =  0.717. 

Furthermore,  '.•  8540  is  halfway  from  8537  to  8543, 
.*.  antilog  2.8540  is  about  halfway  from  antilog  2.8537  to 
antilog  2.8543. 

.'.  antilog  2.8540  is  about  halfway  from  714  to  715. 
.-.  antilog  2.8540  =  714.5. 
Similarly,  to  find  antilog  1.8563. 

antilog  1.8567  =  0.719  1.8563 

antilog  1.8561  =  0.718  1.8561 

6  2 

.-.  antilog  1.8563  =  0.718f  =  0.7183. 

The  interpolation  here  explained  is,  as  stated  on  p.  353, 
merely  a  close  approximation ;  it  cannot  be  depended  upon 
to  give  a  result  beyond  four  significant  figures  except  when 
larger  tables  are  employed. 

This  is  sufficient  in  many  numerical  computations.  £^-g-, 
we  speak  of  the  distance  to  the  sun  as  93,000,000  mi.,  using 
only  two  significant  figures. 


358 


ELEMENTS   OF  ALGEBRA. 


EXERCISES.    CLXII. 

rrom  the  table  find  the  following  : 


1.  antilog  0.3234. 

3.  antilog  2.9193. 

5.  antilog  3.9286. 

7.  antilog  0.8996. 

9.  antilog  3.9320. 

11.  antilog  1.9850. 

13.  antilog  10.5445. 

15.  antilog  0.9485  -  4. 

17.  antilog  0.6120  -  2. 


2.  antilog  2.4271. 

4.  antilog  5.2183. 

6.  antilog  1.7929. 

8.  antilog  4.7834. 

10.  antilog  2.0000. 

12.  antilog  0.7076. 

14.  antilog  3.6987. 

16.  antilog  0.6585  -.6. 

18.  antilog  0.9290  -  3. 


364.  Cologarithms.  In  cases  of  division  by  a  number  n 
it  is  often  more  convenient  to  add  the  logarithm  of  -  than 

to  subtract  the  logarithm  of  n.     The  logarithm  of  -  is  called 
the  cologarithm  of  n. 

•.•  log  -  =  log  1  —  log  n  =  0  —  log  n, 

.'.  colog  w  =  —  log  n. 
Also,   colog  w  —  10  —  log  n  —  10,   often   a  more  convenient  form 
to  use. 

E.g.,   .:  log  6  =  0.7782. 

colog  6  =  -  0.7782. 

This  may  also  be  written  10  -  0.7782  -  10,  or  9.2218  -  10. 

The  object  of  this  is  seen  when  we  consider  the  addition  of  several 
logarithms  and  cologarithms ;  it  is  easier  to  add  if  all  the  mantissas 
are  positive,  subtracting  the  lO's  afterwards. 

In  general,  colog  w  =  lOp—  logw  —  10_p;  that  is,  we  may  use  10, 
20,  or  any  multiple  of  10,  as  may  be  most  convenient. 


LOGARITHMS.  359 

The  cologarithm  can  evidently  be  found  by  mentally  sub- 
tracting each  digit  from  9,  excepting  the  right-hand  signifi- 
cant one  (which  must  be  subtracted  from  10)  and  the  zeros 
following,  and  then  subtracting  10. 

E.g.,  to  find  colog  G178. 

9.     9    9    9  10 
log  6178  =  3.     7     9    0    9 


colog  6178  =  6. 

2    0    9     1-10, 

To  find  colog  41.5. 

9. 

9    9  10    0 

log  41. 5=  1. 

6     18    0 

colog  41. 5  =  8. 

3    8    2    0-10 

To  find  colog  0.013. 

9. 

9    9    9  10 

log  0.013=:^ 

113    9 

colog  0.013  =  11.     8    8    6     1  -  10  =  1.8861. 

In  case  the  characteristic  exceeds  10  but  is  less  than  20, 
colog  n  may  be  written  20  —  log  n  —  20,  and  so  for  other 
cases ;  but  these  cases  are  so  rare  that  they  may  be  neglected 
at  this  time. 

The  advantage  of  using  cologarithms  will  be  apparent 
from  a  single  example : 

317 • 92 
To  find  the  value  of  n  io' 

d17o  •  U.lo 

Using  Cologakithms.  Not  using  Cologarithms. 

log  317=    2.5011  log  317  =  2.5011 

log  92  =    1.9638  log  92  =  1.9638 

colog  6178  =    6.2091  -  10  log(317-92)  =  4.4649 

colog  0. 13  =  10.8861  -  10  log  6178  =  3. 7909 

log  36. 32=    1.5601  log  0.13  =  1.11.39 

log(6178.  0.13)  =  2.9048 

log  (317 -92)  =  4.4649 

317-92         _„,,  log(6178. 0.13)  =  2.9048 

OD.OZ. 


6178-0.13  '     ■  log  36.32=  1.5601 


360  ELEMENTS  OF  ALGEBRA. 

365.  Various  bases.  Thus  far  we  have  considered  loga- 
rithms as  exponents  of  powers  of  10.  Bait  it  is  evident 
that  any  other  base  might  be  taken.  Logarithms  to  the 
base  10,  such  as  we  have  thus  far  considered,  are  sometimes 
called  common  or  Briggs  logarithms,  the  latter  designation 
being  in  honor  of  Henry  Briggs,  who  is  said  to  have 
suggested  this  base  to  Napier. 

If  2  were  the  base,  log  8  would  be  3,  because  2^  =  8. 
Similarly,  log  16  would  be  4,  and  so  on. 

Where  a  different  base  than  10  is  used  (which  is  not  the 
case  in  practical  calculations),  or  where  more  than  one  base 
is  used  in  the  same  discussion,  the  base  is  indicated  by  a 
subscript ;  thus,  logg  32  =  5,  because  2^  =  32. 

366.  Computations  by  logarithms.  A  few  illustrative 
problems  will  now  be  given  covering  the  types  which  the 
student  will  most  frequently  meet.  It  is  urged  that  all 
work  be  neatly  arranged,  since  as  many  errors  arise  from 
failure  in  this  respect  as  from  any  other  single  cause. 

Since  tt  enters  so  frequently  into  computations,  the  follow- 
ing logarithms  will  be  found  useful : 

log  TT  =  0.4971,  log  -  =  1.5029. 


TT 


0.007^ 
1.  Find  the  value  of 


0.03625 


log  0.007  =    0.8451  -    3 

3.  log 0.007  =    2.5353  -    9 

colog  0.03625  =  11.4407  -  10 

13.9760  -  19 

=    0.9760-    6  =  log  0.000009462. 

.-.  9.462  .  10-6  =  Ans. 

It  will  be  noticed  that  the  negative  characteristic  is  less  confusing 
if  written  by  itself  at  the  right. 


LOGARITHMS.  361 

2.  Find  the  value  of  0.09515*. 

*  log  0.09515  =  0.9784 -2. 
•••  the  characteristic  (—  2)  is  not  divisible  by  3,  this  may  be  written 

log0.09515  =  1.9784 -3. 
Then  i  log  0.09515  =  0.6595  -  1  =  log  0.4566. 

.-.  0.4566  =  Ans. 

3.  Given  a,  r,  I,  in  a  geometric  series,  to  find  n.     Compute 
the  value  if  Z  =  256,  a  =  1,  r  =  2. 

1.  From  §  350,  I  =  ar^-K 

2.  .-.  log  Z  =  log  a  +  (n  -  1)  log  r.  §  361 

logr 

log  256  =  2.4082 
logl  =0,  log  2  =  0.3010; 
2.4082  -4-  0.3010  =  8. 

4.  .-.  n  =  8  +  1  =  9. 

,    ^.   ^  ,    •      ,        ^  2.706  •  0.3  •  0.001279 

4.  Find  the  value  oi  ^TrxF^ 

2  706  •  3  •  1  279 
This  may  at  once  be  written  — '         ' 10- 8,  thus  simplifying 

the  characteristics.     Then 

log2.706  =  0.4324 

log  3  =  0.4771 

log  1.279  =  0.1069 

colog  8.609  =  9.0650  -  10 

log  1.206  =  0.0814 

.-.  1.206  .  10-8  =  Ans. 

5.  Given  2^  =  7,  find  x,  the  result  to  be  correct  to  0.01. 

X  log  2  =  log  7. 

....  =  '"81  =  2:^  =  2.81. 
log  2      0.3010 

This  division  might  be  performed  by  finding  the  antilogarithm  of 
(log  0.8451  —  log  0.3010),  a  plan  not  expeditious  in  this  case. 


362  ELEMENTS  OF  ALGEBRA. 

6.  The  weight  of  an  iron  sphere,  specific  gravity  7.8,  is 
14.3  kg.     Find  the  radius. 

■y  =  1 7tr^  •  1  cm^  =  volume  in  cm^. 
.-.  weight  =  f  ;rr8  •  7.8  . 1  g  =  14,300  g. 

r  =  ( )  ,  the  number  of  centimeters  of  radius. 

\  4  7r.  7.8  / 

log  3  =  0.4771 

log  14,300=  4.1553 

colog4=  9.3979-10 

colog  7t  =  9.5029  -  10 

colog7.8=  9.1079  -  10 

3|  2.6411 

log  7.593=  0.8804 

.'.  radius  =  7.593  cm. 

EXERCISES.    CLXIII. 

In  the  following  exercises  give  the  result  to  four  signifi- 
cant figures. 

1.  Find  the  value  of  37 ^V. 

2.  Given  x^  =  x'  :15.     Find  x. 

3.  Find  the  value  of  (32/29)1 


4.  Find  the  value  of  Vtt  •  5.927. 

5.  Find  the  value  of  (5.376 /7r)i 

6.  Find  the  value  of  (37/2939)^*. 

7.  Given  227,600  =  7'*-^     Find  n. 


8.  Find  the  value  of  ^2  ^2  :  VTO. 

9.  Find  the  value  of  (3.64/ 7.985) «. 

10.  Find  the  value  of  v  4.257»  V^OS. 

11.  Find  the  value  of  (1402/3999)-^ 


LOGARITHMS. 


363 


12.  Find  the  value  of  VlOO. 

13.  Find  the  value  of  (22.8  h-  0.09235)^. 

14.  Find  the  value  of  (24.73^  --  31.97*)^. 

15.  Find  the  value  of  (44  •  8.37)^  --  0.227^. 

16.  Find  the  value  of  4  irr'^,  when  r  =  2.06. 

17.  Also  of  I  TTV^. 

18.  Given  x  :  5.127  =  0.325  :  2936.     Find  x. 

19.  Find  the  value  of  *  a^bir,  when  a  =  19.63,  b  =  19.57. 

20.  Given  a,  r,  s,  in  a  geometric  series,  show  that 

_  log  [ct  +  (r  —  1)  6-]  —  log  a 
log  r 
and  compute  the  value  of  n  when  a  =  1,  r  =  2,  s  =  511. 

21.  Also,  given  r,  Z,  s,  show  that 


_  log  I  —  log  [Ir  —  (r  —  1)  5] 
logr 


+  1. 


Compute  the  value  of  n  when  r  =  3,  I  =  729,  s  ==  1092. 

22.    Also,  given  a,  I,  s,  show  that 
log  I  —  log  a 


log  (s -a) -log  (5 -Z) 
Compute  the  value  of  n  when  a  =  3,  I  =  729,  s  =  1092. 

23.  Find  the  values  of  V2,  ^"v^,  ^"v^,  v^,  each  to  3 
decimal  places.  Which  of  these  is  greatest  ?  From  this 
it  may  be  inferred  that  the  value  of  n  that  makes  'Vn 
greatest  is  about  what? 

24.  Solve  the  equation  5^  —  6.  (First  take  the  loga- 
rithm of  each  member.) 

25.  Also  the  equation  Vs  =  10. 


CHAPTER   XX. 

PERMUTATIONS    AND   COMBINATIONS. 

367.  The  different  groups  of  2  things  that  can  be  selected 
from  a  collection  of  3  different  things,  without  reference  to 
their  arrangement,  are  called  the  combinations  of  3  things 
taken  2  at  a  time. 

E.g.^  representing  the  .3  things  by  the  letters  a,  6,  c,  we  can  select 
2  things  in  3  ways,  a&,  ac^  he. 

In  general,  the  different  groups  of  r  things  which  can  be 
selected  from  a  collection  of  n  different  things,  without 
reference  to  their  arrangement,  are  called  the  combinations 
of  n  things  taken  r  at  a  time. 

So  the  combinations  of  the  4  letters  a,  h,  c,  d,  taken  3  at 
a  time,  are  abc,  abd,  acd,  bed ;  taken  2  at  a  time,  ab,  ac,  ad, 
be,  bd,  cd. 

EXERCISES.    CLXIV. 

1.  What  is  the  number  of  combinations  of  5  things 
taken  2  at  a  time  ?     Represent  them  by  letters. 

2.  What  is  the  number  of  combinations  of  5  things 
taken  3  at  a  time  ?     Represent  them  by  letters. 

3.  Write  out  the  combinations  of  the  letters  w,  x,  y,  z, 
taken  4  at  a  time  ;  3  at  a  time ;  2  at  a  time  ;  1  at  a  time. 

4.  How  does  the  number  of  combinations  of  6  things 
taken  2  at  a  time  compare  with  the  number  taken  4  at  a 
time? 

364 


I 


PERMUTATIONS   AND    COMBINATIONS.  365 

368.  The  different  groups  of  2  things  which  can  be 
selected  from  3  things,  varying  the  arrangements  in  every 
possible  manner,  are  called  the  permutations  of  3  things 
taken  2  at  a  time. 

E.g.,  the  permutations  of  the  letters  a,  &,  c,  taken  2  at  a  time,  are 
a6,  6a,  ac,  ca,  be,  cb. 

In  general,  the  different  groups  of  r  things  which  can  be 
selected  from  n  different  things,  varying  the  arrangement 
in  every  possible  manner,  are  called  the  permutations  of  n 
things  taken  r  at  a  time. 

In  all  this  work  the  things  are  supposed  to  be  different,  and  not  to 
be  repeated,  unless  the  contrary  is  stated. 

369.  The  number  of  combinations  of  n  things  taken  r  at 
a  time  is  indicated  by  the  symbol  C".  The  number  of  per- 
mutations of  n  things  taken  ?•  at  a  time  is  indicated  by  the 
symbol  P". 

EXERCISES.    CLXV. 

1.  Show  that  Ft  =  12. 

2.  Show  that  P^  =  2 -PI. 

3.  ShowthatP|  =  2.(7^ 

4.  Find  the  value  of  P^ ;  of  PI 

5.  Show  that  Cl  =  n,  and  C^  =  1. 

6.  Show  that  Pf  =  3,  and  in  general  that  P '{  =  n. 

7.  Using  the  letters  a,  b,  c,  show  that  CI  =  3. 

8.  Write  out  the  permutations  of  the  letters  of  the  word 
time,  taken  all  together. 

9.  Write  out  the  permutations  of  the  letters  a,  h,  c,  d 
taken  2  at  a  time;    3  at  a  time. 


366  ELEMENTS  OF  ALGEBRA. 

370.    Theorem.    The  number  of  permutations  of  ii  different 
things  taken  t  at  a  time  ^s  n  (n  —  1)  (n  —  2)  •  •  •  (n  —  r  +  1). 

Proof.    1.  Since  we  are  to  take  r  things  we  may  suppose 
there  are  r  places  to  be  filled. 
The  first  place  may  be  filled  in  any  one  of  n  ways. 

Thus,  with  a,  b,  c,  d,  we  may  fill  the  first  place  with  a,  6,  c,  or  d. 

2.  For  every  way  of  filling  the  first  place  there  are 
n  —  1  ways  of  filling  the  first  and  second. 

Thus,  if  the  first  place  be  filled  with  a,  we  may  fill  the  first  and 
second  with  ab,  ac,  ad. 

3.  ,'.forn  ways  of  filling  the  first  place  there  are 
n(n  —  1)  ways  of  filling  the  first  two. 

E.g.,  ab,  ac,  ad, 

ba,  be,  bd, 

ca,  cb,  cd, 

da,  db,  dc, 

giving  4  •  3  =3  12  ways  in  all. 

4.  For  every  way  of  filling  the  first  two  places 
there  are  n  —  2  ways  of  filling  the  first,  second, 
and  third. 

Thus,  if  the  first  2  places  be  filled  with  ab,  the  first  3  can  be  filled 
with  abc,  abd,  i.e.,  in  4  —  2  ways. 

5.  .'.  for  n(n  —  1)  ways  of  filling  the  first  two 
places  there  are  n{n  —  l)(n  —  2)  ways  of  fill- 
ing the  first  three. 

E.g.f  abc,      abd,      adc,      adb, 

acb,      acd,      bca,       bed, 
bda,      bdc,       cda,      cdb, 
and  the  same  with  the  first  two  letters  interchanged  in  each. 


PERMUTATIONS  AND   COMBINATIONS.  367 

6.  Similarly,  the  number  taken  4  at  a  time  is 
n(n  —  1)  (n  —  2)  (n  —  3),  and  the  same  reason- 
ing evidently  shows  that  the  number  of  permu- 
tations of  n  things  r  at  a  time  is 


n  (71  -l)(7i-2)---(n-r-  1) 
or         7i(7i  —  l)(7i  —  2)--'(n  —  r  -{- 1). 

Corollary,  i/n  =  r,  P^  =  n(n  —  !)•• -3 -2  •  1.  HeTice, 
the  7iumher  of  permutations  of  n  things  taken  all  together  is 
n(n-l)(n-2)...3.2.1. 

EXERCISES.    CLXVI. 

1.  Find  the  value  of  P^. 

2.  Find  the  value  of  P^. 

3.  Prove  that  Plz\  =  -  Pi. 
n 

4.  Prove  that  Pl  =  P'l.'  P^z^. 

5.  Find  the  value  of  Pf ;  of  Pg.     Prove  this  by  writing 
out  the  permutations  of  the  letters  a,b,c,---. 

6.  Show  from  the  theorem  (§  370)  that  P^.  is  greater  as 
^r  is  greater. 

7.  Show  from  the  corollary  that  P^J  is  the  product  of  all 
I  integers  from  1  to  7i  inclusive. 

8.  Find  the  number  of  permutations  of  the  letters  of 
L  the  word  7iu7nher  taken  all  together. 

f  9.    Find  the  number  of  permutations  of  the  letters  of 

the  word  courage  taken  3  at  a  time  ;   taken  all  together. 

10.  By  writing  out  the  permutations  and  the  combina- 
tions of  the  letters  a,  b,  c,  d,  e,  taken  2  at  a  time,  ascertain 
how  P|  compares  with  C^. 


368  ELEMENTS   OF  ALGEBRA. 

371.  Factorials.     The  product 

n{n  -  1)  (n  -  2)  (n  -  3)-  --S  '2  '1, 
that  is  of  all  integers  from  1  to  n  inclusive,  is  called  fac- 
torial n. 

Thus,  factorial  3  =  1.2-3  =  6, 

4  =  1  .  2  .  3  .  4  =  24,  etc. 

Factorial  n  is  represented  by  several  symbols.  In  writing 
it  is  customary  to  use  \n,  this  being  a  symbol  easily  made. 
In  print,  on  account  of  the  difficulty  of  setting  the  \n,  it  is 
customary  to  use  the  symbol  n !  or  (especially  in  Germany) 
Un. 

n  is  a  Greek  letter  corresponding  to  P,  and  may  be  thought  of  as 
standing  for  product. 

We  shall  use  in  print  only  the  symbol  nl 

372.  It  therefore  appears  that 

(1)  F^  =  n\ 

(2)  P"-  ^K^^-l)(^-2)---3-2-l    _       nl 


(n-r){n-r-l)...3-21       (n  -  7-)l 


EXERCISES.    CLXVII. 
10' 

1.   Showthat  P\o  =  — -'.         2.    Show  that  5!  =  120. 

3.    Find  the  value  of  ^.      4.    Also  of  ^  •  ^  •  ^- 
o !  10 !    D !   21 

5.  Prove  that  nl  =  n(n  -  1)  (n  -  2)  -  (n  -  3)1 

6.  Prove  that  (niy  =  n\n  -  iy(n  -  2y-  ■  ■3'' -2^ -1. 

7.  In  how  many  ways  can  10  persons  be  placed  in  a 
row? 


PERMUTATIONS   AND   COMBINATIONS.  369 

373.  Theorem.  The  number  of  permutations  of  n  differ- 
ent thi7ujs  taken  v  at  a  time,  when  each  of  the  n  things  may 
be  repeated,  is  n^. 

Proof.    After  the  first  place  has  been  filled,  the  second 
can  be  filled  in  n  ways,  since  repetition  is  allowed. 
So  for  the  subsequent  places. 
Hence,  instead  of  having 

P-  =  n{n  -  1)  {n  -  2)  ■  •  -{n  -  r  +  1), 

we  have  n-n-n-  •  -n  =  n^. 

EXERCISES.    CLXVIII. 

1.  Find  the  value  of  P\,  repetitions  being  allowed. 

2.  Find  the  value  of  PI,  repetitions  being  allowed. 

3.  How  many  numbers  are  there  containing  4  digits  ? 

4.  How  many  ways  are  there  of  selecting  3  numbers 
from  50  on  a  combination  lock,  repetitions  being  allowed  ? 

5.  How  many  ways  are  there  of  selecting  3  numbers 
from  10  on  a  combination  lock,  repetitions  being  allowed  ? 

6.  Show  that  P^,  repetitions  being  allowed,  is  n"".  From 
this  tell  how  many  9-figure  numbers  are  possible,  all  zeros 
being  excluded. 

7.  From  ex.  6,  how  many  10-figure  numbers  are  possible, 
zeros  being  admitted  except  in  the  highest  order. 

8.  How  many  possible  integral  numbers  can  be  formed 
from  the  digits  1,  2,  3,  4,  or  any  of  them,  repetitions  of  the 
digits  being  allowed  ? 

9.  The  chance  of  guessing  correctly,  the  first  time,  the 
three  numbers  on  which  a  combination  lock  of  100  numbers 
is  set,  is  1  out  of  how  many  ? 


370  ELEMENTS  OF  ALGEBRA. 

374.    Theorem.     The  number  of  combinations  of  n  different 
things  taken  t:  at  a  time  is 

n(n-l)(n-2)-.-(n-  r  +  1) 
r! 
Proof.    1.  For  each  combination  of  r  tilings  there  are  r\ 
permutations. 

2.  .'.  for  C"  combinations  there  are  C^  x  r\  per- 
mutations. 

3.  But  it  has  been   shown   that  this  number  of 
permutations  is 

n{n  -  l){n  -  1)-  •  -{n  -  r  +  1).  §  370 

4.  .-.  6':^  X  r!  =  ?^(7^-l)(/i-2).••(7^-r  +  l), 
and  C«  =  ^K^-l)(^-2)---(r^-r  +  l) 

Corollaries.    1.    C"  =  P^'^/rl 
n\ 


2.   Cl 


r\  (n  —  r)\ 

For  we  may  multiply  both  terms  of  the  fraction 
n{n  —  1)  {n  —  2)-  •  ■  {n  —  r  +  1) 
r\ 
hj  {n-r)\,  giving 

n{n-l){n-2)---{n-r  +  l){n-r){n-r-l)-.-^.2-l 
r\(n  —  7')l 

which  equals  — '■ 

r\{n  —  r)\ 

This  is  a  more  convenient  formula  to  write  and  to  carry  in  mind. 
Practically,  of  course,  it  gives  the  same  result  as  the  other.     E.g.^ 

By  the  theorem,  0%  =  5jAl5  • 

by  the  corollary,  C|  =  ^'^'^"^•^ . 

3.21-2.1 


PERMUTATIONS   AND   COMBINATIONS.  371 

EXERCISES.    CLXIX. 

1.  If  Pi  =  3,628,800,  find  n. 

2.  Find  the  values  of  P^;  of  P'^;  of  CI 

3.  If  P%  =  bQ>,  find  n,  and  explain  why  there  should  be 
two  results. 

4.  In  how  many  ways  can  3  persons  be  selected  from  a 
class  of  20  ? 

5.  In  how  many  ways  can  the  letters  of  the  word  cat 
be  arranged  ? 

6.  Prove  that  C;?  =  C„!!.^,  by  substituting  in  the  formula 
of  §  374,  cor.  2. 

7.  What  is  the  number  of  combinations  of  20  things 
taken  5  at  a  time  ? 

8.  In  how  many   ways    can  the   letters  of   the  word 
number  be  arranged  ? 

9.  How  many  numbers  can  be  formed  by  taking  4  out 
of  the  5  digits  1,  2,  3,  4,  5  ? 

10.  How  many  triangles  are  formed  from  4  lines,  each  of 
which  intersects  the  other  3  ? 

11.  How  many  changes  can  be  rung  with  a  peal  of  7 
bells,  a  particular  one  always  being  last  ? 

12.  In   how  many  ways  may  the  letters  of   the  word 
united  be  arranged,  taken  all  at  a  time? 

13.  How  many  changes  can  be  rung  with  a  peal  of  5 
bells,  using  each  bell  once  in  each  change  ? 

14.  In  how  many  ways  can  a  consonant  and  a  vowel  be 
chosen  out  of  the  letters  of  the  word  numbers  ? 


372  ELEMENTS   OF   ALGEBRA. 

15.  How  many  numbers  between  2000  and  5000  have 
the  hundreds  figure  7  and  are  divisible  by  2  ? 

16.  In  how  many  ways  may  the  letters  of  the  word 
rate  be  arranged,  taken  any  number  at  a  time  ? 

17.  In  how  many  ways  can  5  persons  be  seated  about 
a  circular  table,  one  of  them  always  occupying  the  same 
place  ? 

18.  How  many  different  arrangements  (permutations) 
can  be  made  by  taking  5  of  the  letters  of  the  word  tri- 
angle ? 

19.  On  an  examination  15  questions  are  given,  of  which 
the  student  has  a  choice  of  10.  In  how  many  w^ays  may  he 
make  his  selection  ? 

20.  How  many  different  arrangements  can  be  made  of 
the  letters  of  the  word  algebra,  it  being  noted  that  two  of 
the  letters  are  alike  ? 

21.  There  are  four  points  in  a  plane,  no  three  being  in 
the  same  straight  line.  How  many  straight  lines  can  be 
drawn  connecting  two  points  ? 

22.  How  many  different  signals  can  be  made  with  5 
different  flags,  displayed  on  a  staff  3  at  a  time  ?  4  at  a 
time  ?  2  at  a  time  ?  altogether  ?  any  number  at  a  time  ? 

23.  Suppose  a  telegraphic  system  consists  of  two  signs, 
a  dot  and  a  dash ;  how  many  letters  can  be  represented  by 
these  signs  taken  1  at  a  time  ?  2  at  a  time  ?  3  at  a  time  ? 
4  at  a  time? 

24.  Prove  that  the  number  of  permutations  of  7i  different 
things  taken  r  at  a  time  is  n  —  r  +  1  times  the  number  of 
permutations  of  the  n  things  taken  ?'  —  1  at  a  time. 


CHAPTER   XXI. 
THE   BINOMIAL   TPIEOREM. 

375.  The  binomial  theorem  is  stated  in  §  80,  and  a  proof, 
which  may  be  used  in  connection  with  that  section,  is  given 
in  Appendix  I. 

It  is  now  proposed  to  consider  this  theorem  in  the  light 
of  Chapter  XX. 

376.  Theorem.  If  the  binoviial  a  +  b  ^s  raised  to  the  nth 
power,  n  integral  and  jjositive,  the  result  is  expressed  by  the 
formula 

(x  +  a)"  =  X"  +  C?  x"-ia  +  C^  x"-2a2 

+  Cjx"-'V  H C^^Lixa"-^  +  a". 

Proof.    1.  By  multiplication  we  know  that 
{x  +  a){x-\-  h) 

^  x"^  -\- {a  -{- h)  X  -\-  ab, 
{x  +  a){x-{-  b)  {x  +  c) 

^x^-\-(a  +  b-\-c)x^+  (ab  -\-bc-\-  ca)  x  +  abc, 
{x  -{-  a){x  +  b)  {x  +  c){x  +  d) 
=  x^-\-{a  +  b  +  G  +  d)x^ 

4-  (ab  +  ac  -{-  ad  -\-  bo  -\-  bd  -\-  cd)  x^ 
+  (abc  +  abd  -\-  acd  +  bed)  x  +  abed. 

There  is  evidently  a  law  running  through  all 
these  expansions,  relating  to  the  exponents  and 
the  coefficients  of  x. 
373 


374  ELEMENTS  OF  ALGEBRA. 

2.  We  might  infer  from  step  1  that  if  there  were 
n  factors,  the  product  would  have  for  the  coef- 
ficient 

of  ic",      1 ; 

of  cc"~"^,  a  -\-  b  -{-  c-'-n; 

of  ic""^,  the  combinations  of  the  letters  a,b,---n, 
taken  2  at  a  time ; 

of  x""-^,  the  combinations  of  these  letters  taken 
3  at  a  time ; 

of  X,        the  combinations  of  these  letters  taken 
7i  —  1  at  a  time. 

3.  This  inference  is  correct ;  for  the  term  con- 
taining a;"  can  be  formed  only  by  taking  the 
product  of  the  x's  in  all  the  factors,  and  hence 
its  coefficient  is  1. 

The  terms  containing  ic"-^  can  be  formed  only 
by  multiplying  the  ic's  in  all  but  one  factor  by 
the  other  letter  in  that  factor;  hence  the  x''-'^ 

term  will  have  for  its  coefficient  (a  +  b  -\ 7i). 

The  terms  containing  a;"-^  q^^^  i^q  formed  only 
by  multiplying  the  x's  in  all  but  2  factors  by 
the  other  letters  in  those  factors,  i.e.,  by  a  and 
b,  a  and  c,  a  and  d,  etc. ;  hence  the  cc"~^  term  will 

have  for  its  coefficient  (ab  -{-  ac  -\-  ad  -\ ). 

The  reasoning  is  evidently  general  for  the  rest 
of  the  coefficients. 

4.  If,  now,  we  let  a  =  b  —  G  =  ---  =  n,  we   have 

(x  -f  ay  =  x''+  C\x''-\i  +  C^x«-2a2 

+  C^ic"- V  -\ 

+  C^l^xa''-^  +  ft". 


THE    BINOMIAL   THEOREM.  375 

As  stated  in  §  246,  the  binomial  theorem  is  true  whether 
n  is  positive  or  negative,  integral  or  fractional.  While  the 
proof  of  this  fact  cannot  satisfactorily  be  presented  without 
the  differential  calculus,  the  fact  itself  should  be  recog- 
nized. 

The  following  exercises  will  serve  to  recall  the  applica- 
tion of  the  theorem,  although  they  do  not  differ  materially 
from  those  already  met  by  the  student  in  the  exercises 
following  §§  80,  246. 

Illustrative  problems.  1.  Required  the  square  root  of 
1  -\-  X  to  3  terms. 

1.  •.•      (a  +  6)»  =  a»  +  na«  - 16  +  ^'^^^?— -^  a" -262  4- ..., 

2.  .-.      (1  +  x)^  =  1^  +  i  •  1~  ^  •  X  +  ^  ^^  ~  ^^  •  1~ ^  •  x^  +  •  •  • 

=  l+iX-lx2  +  .... 


2.  Expand  to  4  terms  {a  —  2  b)-^. 

1.  V  (x  +  ?/)«  =  x'*  +  nx'^-^y  +  ^^^~    ' x'^-^y^ 

n{n-l){n-2)   ^    „  „ 
2-3  .      ^   ^ 

2.  .-.  (a-2  6)-3  =  a-'^  +  (-3)a-4(-2  6) 

=  a-3  +  6  a-46  +  24  a-''^b^  +  80  a-%^  + 


3.  Expand  to  3  terms  (1  +  x)   *. 
As  above,  (1  +  x)-i  =  1  +  {-  i)x  +  (~  ^)  ("  ^  "  ^)  3.2  4. 
=  1  ~ix-h^%x^---. 


376  ELEMENTS   OF  ALGEBRA. 

EXERCISES.    CLXX, 

Expand  the  following  binomials  : 

1.   (cc  +  5)^  2.   (x^-2af. 

(^       a;\i«  (a       h\ 

5.    (40 +  1)^  6.    {?>a-\hy. 

7.  (1  +  xy,  to  4  terms. 

8.  {a  +  V)^,  to  4  terms. 


9.  -\  d^  —  £c^,  to  3  terms. 

10.  (1  +  x)~^,  to  4  terms. 

11.  (1  —  2  a)*,  to  4  terms. 

12.  (3  a:  -  2?/)^,  to  4  terms. 

13.  v'.Si  =  (32  -  1)^,  to  3  terms. 

14    -—==  =  (1  +  ic)~  ,  to  4  terms. 
'    Vl  +  a; 

15.  (1  —  x)~'^,  to  5  terms,  checking  by  performing  the 
division  z 


16.  (1  —  x)~^,  to  5  terms,  checking  by  performing  the 

division t: o' 

1  -  2  a:  +  ic^ 

17.  (1  -{- x)~^,  to  5  terms,  checking  by  performing  the 

division -• 

1  +  2  a:  +  a;2 


APPENDIX. 


I.     PROOF  OF   THE  BINOMIAL  THEOREM   FOR   POSITIVE 
INTEGRAL  EXPONENTS  (p.  57). 


If  n  is  a  positive  integer 


{a  +  Z»)»  =  a"  +  na''-^h  + 


+ 


n{n-V)  {n  -  2) 
23 


-3^3  _^ 


Proof.     1.    The  law  is  evidently  true  for  the  2d  power,  for  {a  +  h)^ 
=  a2  +  2  a6  +  h'^,  or,  as  the  theorem  says, 

=  a25o  +  2  aifti  +  a^62.  §  69 

2.  It  is  also  true  for  the  3d  power,  for  (a  +  6)3  =  a^  +  3  a'^h  +  3  a&2 
+  63,  or,  as  the  theorem  says, 

=  aPW  +  3  a26i  +  3  ai62  +  ^0^3.  §  69 

3.  Now  if  it  were  true  for  the  fcth  power  we  should  have  (a  +  6)* 

and  if  this  were  multiplied  by  a  +  6  we  should  have 

4.  (a  +  6)^-  +  i 


=  #+1  +  A: 

+  1 


#6 


fc(fc-l) 


afc-162  + 


A;  (fc  -  1)  (A;  -  2) 


2.3 

k{k-\)  • 


a*-263  + 


a^+i  +  (fc  +  i)a%  4-  ^A±il^a^-i62 


(fc  +  l)fc(fc-l)^,_ 


2-3 


■253 


5.    But  here  we  see  that  i/  the  theorem  were  true  for  any  power ^  as 
the  fcth,  it  would  be  true  for  the  next  higher  power,  as  the  {k  +  l)th. 

0.    But  the  theorem  is  true  for  the  3d  power  (step  2),  and 

.-.  it  is  true  for  the  (3  +  l)th  or  4th  power,  by  step  5 ; 

.-.    "  "       "    (4  +  l)th  "  5th      "         "       " 

and  so  on  for  all  integral  powers. 

377 


378  ELEMENTS  OF  ALGEBRA. 

IL     SYNTHETIC  DIVISION   (p.  67). 

If  the  divisor  is  a  binomial  of  the  first  degree,  there  is 
often  a  considerable  gain  by  resorting  to  a  form  of  division 
known  as  synthetic. 

The  process  is  best  understood  by  following  the  solution 
of  a  problem. 

Required  the  quotient  of  £c^  —  3  cc^  +  3  ic  +  4  by  x  —  1. 
The  ordinary  long  form  would  be  as  follows,  the  heavy 
numerals  being  the  ones  reserved  in  the  synthetic  form 
given  below : 


x^ 

-2x 

+  1 

x^ 
x^ 

-3x^ 

-Ix^ 

-\-3x 

+  4 

-2x^ 

H-2a' 

a;  +  4 

X-  1 


o  rem. 


This  may  be  abridged  by  writing  the  quotient  below,  as 
follows : 


3x^  +3x  +4 
lx%-\-2x,-  1 


2x  -\-l\        5  rem. 


Here  the  first  term  of  the  quotient,  x"^,  is  multiplied  by 
—  1,  this  product  subtracted  from  —  3x'^  and  the  remainder 
immediately  divided  by  x  to  get  the  next  term,  —  2x,  and 
so  on. 

Since  it  is  easier  to  add  than  to  subtract,  it  is  usual  to 
change  the  sign  of  the  second  term  of  the  divisor  and  add. 
Doing  this,  and  detaching  the  coefficients,  we  have  the 
common  form  for  synthetic  division,  as  follows : 


1 

+  1 


APPENDIX.  379 

1-3+3+4 
1-2      1 


1  —  2  +  1 ;     5  rem. 

Check.     Let  x  =  2.     Then  (8 -12 +  6  +  4- 5)  ^1=4-4  +  1. 

In  case  any  powers  of  a  letter  are  wanting  in  arranging 
according  to  descending  powers  of  that  letter,  zero  coeffi- 
cients should  be  introduced  as  usual. 

EXERCISES.    CLXXI. 

Perform  the  following  divisions  by  the  synthetic  process, 
detaching  the  coefficients,  and  checking  in  the  usual  way. 

1.  a^  +  b^  hj  a  +  h. 

2.  x^  —  y^  by  x  —  y. 

3.  a^  —  4  <x  +  3  by  a  —  1. 

4.  £c^  —  2  x^  +  1  by  a;  +  1. 

5.  1  +  ic  +  x^  +  ic^  by  1  +  a;. 

6.  x^  -  29  X  +  190  by  x  -  10. 

7.  ic^  +  3  x'^a  —  4  xa^  by  x  —  a, 

8.  x^  +  2  x^  —  4  X  +  1  by  x  —  1. 

9.  x^  -  3  x^  +  2  X  +  6  by  X  +  1. 

10.  x«  +  3x2  +  3x  +  28  by  x  +  4. 

11.  5  x'^  +  4  x^  +  3  ic^  +  2  £c  +  1  by  X  +  1. 

12.  a*  -  4  a^^»  +  6  a%^  -  4  a6»  +  ^»*  by  a-  h. 

13.  x^  —  10  ic^  +  9  by  X  +  3  ;    also  by  x  —  3. 

14.  3  x^  -  2  x^  -  7  X  -  2  by  X  +  1 ;    also  by  x  -  2. 

15.  2x2  +  3x?/ —  2?/2  by  X  +  2?/;    ^Iso  by  y  —  2x. 


380  ELEMENTS   OP  ALGEBRA. 

III.     THE  APPLICATIONS    OF   HOMOGENEITY,    SYMMETRY, 
AND   CYCLO-SYMMETRY  (p.  73). 

The  applications  of  homogeneity,  symmetry,  and  cyclo- 
symmetry  are  very  extensive  and  they  materially  simplify 
the  study  of  algebra.  The  principle  which  lies  at  the  foun- 
dation of  these  applications  is  as  follows  : 

If  two  algebraic  expressions  are  homogeneous,  sijmmetric, 
or  cyclic,  their  sum,  difference,  product,  or  quotient  is  also 
homogeneous,  symmetric,  or  cyclic,  respectively. 

The  truth  of  this  principle  follows  from  the  definitions 
and  from  previous  proofs.  E-g-,  by  the  law  of  the  forma- 
tion of  the  product  of  two  polynomials  it  appears  that  each 
term  of  one  factor  is  multiplied  by  each  term  of  the  other ; 
hence,  if  one  factor  is  homogeneous  and  of  the  third  degree 
and  the  other  is  homogeneous  and  of  the  second  degree,  then 
the  product  must  be  homogeneous  and  of  the  fifth  degree. 

The  converse  is  not  necessarily  true.  E.g.,  the  sum  of 
two  non-symmetric  expressions  may  be  symmetric,  as  the 
sum  a^  +  h"^  -\-  G  and  c(c  —  1). 

These  considerations  suggest  some  valuable  checks  on  the 
four  fundamental  operations.  Since  algebraic  expressions 
are  often  homogeneous,  symmetric,  or  cyclic,  these  checks 
will  be  of  service  throughout  the  study  of  the  subject. 

E.g.,  the  product  of  x'^  -\-  y^  and  x  -\-  y  \s  x^  -{■  xy^  +  x'^y  +  y^.  This 
may  be  checked  by  arbitrary  values,  or  by  noticing  that  the  product 
must  be  homogeneous,  of  the  third  degree,  and  symmetric  as  to  x  and  y. 

In  the  same  way  the  square  root  of  a^  +  6^  +  c^  +  2  a&  +  2  6c  +  2  ca 
must  be  symmetric,  the  product  of  (a  —  6)  (6  —  c){c  —  a)  must  be  cyclic 
and  the  quotient  of  27  a^h^  -|-  c^  by  3  a6  -|-  c  must  be  symmetric  as  to  a 
and  6 ;  otherwise  there  must  be  an  error  in  the  operation. 

It  so  happens  that  many  of  the  expressions  dealt  with  in  higher 
algebra  are,  or  can  be  made,  symmetric  or  homogeneous,  or  both,  and 
hence  the  value  of  these  checks  becomes  the  more  apparent  as  the 
student  progresses  in  mathematics. 


APPENDIX.  381 

EXERCISES.    CLXXII. 

Perform  the  operations  here  indicated,  checking  each  by 
substituting  arbitrary  values  and  also  by  (1)  homogeneity, 
(2)  symmetry,  or  (3)  cyclo-symmetry,  as  seems  best. 

1.  (x^J^i/){x^-xY  +  y^)- 

2.  (2x  +  y-z){2x-ij  +  z). 

3.  (81  a'^h'^  -  256  c'')  --  (3  a^>  +  4c). 

4.  (a  -\-  h  -\-  c)  (he  -{-  ca  -\-  ah)  —  abc. 

5.  (a;*  +  x^-i/  +  if)  -^  (x^  +  2/^  +  ^y)- 

6.  —  (a  —  b)  (b  —  c)(c  —  a)  (a  +  b  -{-  c). 

7.  a\b-c)-irb'(c-a)-\-c\a-b). 

8.  a^  (b  —  c)-{-  b^  (c  —  a)  +  c\a  —  b). 

9.  —(a  —  b)  (b  —  c)  (c  —  a). 

10.  a''(b  -  c)-\-b''{c  -  a)-{-  c'ici  -h). 

11.  {a^J^b^  +  1-^  ah)  --(«  +  &  +  1). 

12.  (x  -  y)  (x^  +  xhj  +  xY  +  xy'  +  y')- 

13.  (£C^  -  18  xY  +  2/')  ^  (^^  -  .y'  +  4  ^2/)- 

14.  (a;2  _^  ^2  ^  ^2  _  ^^  _  ^^  _  ^^>)  ^^  _l_  ^  _^  ^^^ 

15.  (a;  +  2/  -  2^)2  +  (y  +  ^  -  2a^)2  +  (^  +  X  -2yy. 

16.  (7^  -  2  Z  -  37/z)2  +  (Z  -  2  m  -  3  A;)2  +  (m  -  2  Z;  -  3  Z)^. 

17.  (^^2  _  ^2  _  ^2  _^  ^2  _^  2  ^c  -  2  acZ)  -f-  (a  +  ^»  -  c  -  d). 

18.  (i?  +  2-  +  r)'  +  (i>'  -  2'  -  r)«  +  (2'  -  r  -  py 

19.  (a;  +  ?/  +  ^)^  -  (?/  +  ^  -  cr)^  -  (^  +  ir  -  ?/)« 

-(x  +  y-  zy. 


382  ELEMENTS   OF   ALGEBRA. 

Symbolism  of  symmetric  expressions.  Since  the  terms  of 
a  symmetric  expression  are  so  closely  related  in  form,  it  is 
often  necessary  to  write  only  the  types  of  these  terms. 

E.g.,  if  a  trinomial  is  symmetric  as  to  a,  b,  and  e,  and  if 
one  term  is  ab,  the  others  are  at  once  known  to  be  be  and  ca. 
The  term  ab  is  therefore  called  a  type  term. 

The  Greek  letter  IS  (sigma,  our  S)  is  used  to  mean  "  the 
sum  of  all  expressions  of  the  type  •  •  • ." 

E.g.,  in  f(x,  y,  z),  %xhj  means  "the  sum  of  all  expres- 
sions of  the  form  xhj "  which  can  be  made  from  the  three 
given  letters. 

That  is,  ^xhj  =  x'^y  +  x^z  +  y^x  +  y^z  +  z'^x  +  z'^y.  This 
polynomial  is  called  the  expansio7i  of  ^x^y. 

If  these  same  three  letters  are  under  discussion, 
^x^  =  x^  +  y^  +  z%  but  (2ic)2  =  (x  +  y  +  zf. 

In  case  of  any  doubt,  the  letters  under  discussion  are 
written  below  the  %,  thus  : 

2  (a  +  6)  =  (a  +  6)  +  (6  +  c)  -\- {c -\- a) 

ahc 

S  (x2  +  y)  =  (x2  +  y)  +  (2/2  +  x). 

xy 

If  an  expression  is  known  to  be  cyclic,  %  has  a  slightly 
different  meaning.  It  then  stands  for  "  the  sum  of  expres- 
sions of  this  type,  ivhich  can  be  formed  by  a  cyclic  hder- 
change  of  the  letters.''^ 

E.g.,  if  only  cyclic  expressions  involving  three  letters  are 
under  discussion, 

%(a-b)  =  {a-b)  +  {b-c)  +  {G-a), 
instead  of 

{a-b)  +  {b-a)  +  {b-c)  +  {c-b)  +  (c-a)  +  {a  -  c) ; 
and 

%a{b  +  c-2  af  =  a{b  +  c  -  2  af  +-b{c  +  a  -  2bY 
■\-c{a  +  b-2cy. 


APPENDIX.  383 

EXERCISES.    CLXXIII. 

Expand  the  expressions  in  exs.  1-8. 

1.    ^xy,  where  only  x,  y,  z  are  involved. 


I 


2.    %a%, 

u 

a,  b 

a 

3.    %{a^hY 

ii 

a,  b,  c 

(( 

4.    ^aJ)" 

u 

u 

(( 

5.    %xhfz. 

xyz 

6.    %a} 

ahc 

'  -  3  abc. 

7.    %a^-\-2^ab. 

8.    ^a^ 

+  3  %a%  +  6  abc. 

In  cyclic  functions  involving  only  a,  b,  e,  what  is  the 
expansion  of  the  expressions  in  exs.  9-14  ? 

9.    ^a(b  +  c).  10.    ^a'^(b-c). 

11.    :Sa'(^>' -c«).  12.    %a%\a-b). 

13.    2a'(a-b  +  c).  14.    [^C^^  -  c)^(^' +  c  -  2  a). 

Show  that  the  following  identities  are  true,  by  expanding 
both  members.  Those  involving  negative  signs  are  cyclic. 
Except  as  stated  to  the  contrary,  only  a,  b,  c  are  involved. 

15.    ^a{b-c)^0.  16.    (:^ay-^a^  =  2:^db. 

17.    (^ay=^a^-{-:$2ab.         18.    ^[(^ay  -  ^a^2  =  %ab. 

a  •■■  d  a  •■■e 

19.  ■%(a-b)(a  +  b  -c)^0. 

20.  '$(a-by=3(a-b)(b-c)(c-a). 

21.  %a\b  -c)  =  -{a-b){b-c){c-  a). 

22.  %ab  (a  —  b)  =  —(a  —  b)(b  —  c)  (c  —  a). 

23.  (Sa)  (^ab)  -  aZ-c  =  (a  +  ^)  (Z»  +  c)  (c  +  a). 
24.    (2a)  {^a^  +  2abc={a  +  b)  (b  +  c)  (c  +  a)  +  2a^ 


384  ELEMENTS  OF  ALGEBRA. 

Illustrative  problems.  The  preceding  principles  render  it 
easy  to  simplify  certain  expansions  which  would  otherwise 
require  considerable  labor.  The  process  will  be  understood 
from  a  few  problems.       ^ 

1.  Expand  {a  +  b  -\-cy. 

1.  The  expression  is  symmetric  and  homogeneous. 

2.  .-.  the  expanded  form  contains  only  the  types  a^,  a&,  with  numer- 
ical coefficients. 

3.  ,-.  it  is  of  the  form  ni'Za'^  +  nSa6,  where  we  have  to  determine 
m  and  n. 


4.  Considering  the  expression  as  a  binomial,  (a  +  6  +  c)2,  we  shall 
evidently  have  cC^  -\-  2  ah  -\- Jfi  -{■  some  terms  which  do  not  contain  cC^ 
or  ah. 

5.  .-.  the  coefficients  of  the  type  a^  are  all  1,  and  those  of  the  type 
ah  are  all  2.     .-.  m  =  1,  w  =  2. 

6.  .-.  the  result  is  Sa^  ^  2  Sa6,  or  a^  +  h^  -\-  c^  -^  2  (ah  -\- ca -\-  he). 
Check.     Let  a  =  6  =  c  =  1.     Then 

32  =  12  +  12  +  12  +  2  (1  +  1  +  1)  =  9. 

2.  Simplify 

{a  +  b  +  cf  ^{a  +  h  -  cf  +  {h  +  c  -  ay  +  {c  -\-  a  -  bf. 

1.  As  in  problem  1,  the  types  are  a2,  a6,  and  the  expanded  form  is 
m2a2  +  nliah,  where  we  have  to  determine  m  and  n. 

2.  In  the  four. trinomials  we  have  a2,  a2,  (—  a)2,  (—  a)2,  or  4a2,  as 
shown  in  problem  1.     .-.  m  =  4. 

3.  Also  2  a&,  2  ah,  -  2  ah,  -  2  a6,  or  0  •  ah.     .•.n  =  0. 

4.  .-.  the  result  is  4  Sa2,  or  4  {a^  +  h^  +  c2). 

Check.     Let  a  =  6  =  c  =  1.     Then 

32  +  12  +  1-2  ^.  12  ^  4  (12  +  12  +  12)  =  12. 

This  particular  problem  is  so  simple  that  there  is  no  great  gain  by 
using  the  S  symbolism. 


APPENDIX.  385 

3.  Expand  (iSa)^  where  ^a  =  a  +  h  +  c  +  d-\-e^ . 

1.  What  can  be  said  of  (Sa)^  as  to  symmetry  ?   homogeneity  ? 

2.  .-.  the  expanded  form  contains  only  what  types  ? 

3.  .-.  it  is  of  what  form,  and  what  coefficients  are  to  be  determined  ? 
(See  problem  1.) 

4.  What  are  these  coefficients  in  the  expansion  of  (a  +  h)^  ? 

5.  Will  the  addition  of  other  letters,  as  c  +  cZ  +  e  +  •  •  • ,  affect  these 
coefficients  of  cfi  and  ah  ? 

6.  .-.  what  values  have  the  coefficients  m  and  w,  and  what  is  the 
result  ? 

4.  Expand  (Sa)^,  where  %a^a-{-h-[-c-\-d-\-e-\----. 

1.    The  types  are  evidently  of  the  third  degree,  and  therefore  must 
be  a3,  cfih,  abc.     (Why  ?) 


2.  In  expanding  (a  +  b  +  c)^,  we  have  (§  69) 

'a+V^  +  8  a  +  b^  ■  c  +  3a  +  6  •  c^  +  c^, 
in  which  the  coefficient  of  a^  is  evidently  1,  of  a^b  is  3,  and  of  abc 
(found  only  in  3  a  +  b^  •  c)  is  6. 

3.  The  addition  of  other  letters,  d  +  e  +  ■  •  •,  will  not  affect  the 
coefficients  of  a^,  a^b,  or  abc. 

4.  .-.  (Sa)3  =  2a3  +  3  Sa26  +  6  Sa6c. 


5.  Expand  (x  -{-  y  -{-  zy  —(y  -\-  z  —  xy  —  (z  +  x  —  yY 
-{x  +  y-  zy. 

1.  What  are  the  types  ? 

2.  •.•  we  have  x^,  —  ( —  x)^,  —  x'\  —  x^,  what  is  the  coefficient  of  Sx^  ? 

3.  ■.•  we  have  3x2?/,  —  3x2?/,  _  ^_  Sx^y),  —  3x2?/,  what  is  the  coef- 
ficient of  2x2?/  ? 

4.  •.•  we  have  6xyz  (as  in  problem  4),  —  (—  6xyz),   —  (—  Gxyz), 
—  (—  6x2/2),  what  is  the  coefficient  of  Xxyz  ? 

5.  .-.  the  result  is  24x?/z. 

Check.     Let  x  =  ?/  =  z  =  1.     Then,  etc. 


386  ELEMENTS   OF   ALGEBRA. 

EXERCISES.    CLXXIV. 

]§  is  limited  to  three  letters  in  each  of   the  following 
exercises,  except  as  otherwise  indicated. 

1.    Expand  (%ay.  2.    Expand  (^ay. 

a  ■■•  d 

3.  Show  that,  if  5a  =  0,  (^ay  =  4  (^aby. 

4.  Show  that  %a  ■  (^a^  -  -^ab)  =  %a^  —  S  abc. 

5.  Show  that,  if  2a  =  0,  :S(a  +  by  +  ^a^  =  0. 

6.  Show  that  (a  +  b)  (b  +  c)  {g  +  a)  =  %a%-^2  abc. 

7.  Simplify  {a  -  b  -  cy +  (b  -  a  -  cy +  {c  -  a  -  by. 

8.  Show  that  %x  •  (Sic  -  2  cc)  •  {%x  -  2ij)  -  {%x  -  2z) 

=  2  %xY  -  ^x\ 

9.  Simplify  (a  -  2b  -  3  cy -\- (b  -  2c  -  Say 

^(^c-2a-Sby. 

10.  Show  that  (—  a  +  b  +  c){a  —  b  +  c)(a  -\-b  —  c) 

=  '%a^(b  +  c)-^a^-2abc. 

11.  Show  that  %{a-b)  =  0. 

12.  Show  that  {a-[-b  -[-c){—  a  +  b  -\-c)(a  —  b  -\-c) 

{a  +  b-c)  =  ^2  a'b''  -  ^a". 

13.  Show  that  {a  +  &)(&  +  c)  (c  +  a)  =  %ab'^  +  2  a^»c. 

14.  Show  that  2a  •  ^a^  ^  a&  (a  +  6)  +  be  (b  +  c) 

+  ca  (c  +  a)  +  2a^. 

15.  Show  that  2a  •  ^ab  =  a'^{b  +  c)  +  b'^  (c  +  a) 

+  c^  (a  +  ^)  +  3  a^>c. 

16.  Show  that  (2a  -  2  a)  (2a  -  2  5)  (2a  -2  c) 

=  a2(5  +  c)  +  ^'(c  +  a)  +  c2(a  +  ^)  -  ^a^  -  2  a6c. 


APPENDIX.  387 

IV.     APPLICATION  OF  THE  LAWS   OF   SYMMETRY  AND 
HOMOGENEITY  TO  FACTORING  (p.  88). 

Since  many  of  the  expressions  in  mathematics  are  sym- 
metric or  homogeneous  or  both,  the  application  of  the  laws 
of  symmetry  and  homogeneity  is  of  great  importance. 

E.g.,  to  factor  ac^  +  ha?  +  c62  _  ah"^  —  bc^  —  ca^,  it  should  be  noticed 
that 

1.  It  is  homogeneous,  of  the  third  degree,  and  cyclic. 

2.  .-.  either  it  has  3  linear  factors,  a  —  b  being  one  (why  ?),  or  else 
it  has  1  linear  factor,  a  +  b  +  c  (why  ?)  and  1  quadratic  factor.    (Why  ?) 

3.  And  •.•  it  vanishes  for  a  =  6,  .-.  a  —  6  is  a  factor,  and  .-.b  —  c  and 
c-a.     (Why  ?) 

4.  There  are  no  more  literal  factors  (why  ?),  but  there  may  be  a 
numerical  factor  n. 

6.    Then  ac^  +  ba^  +  cb'^  -  a¥  -  be?  -ca?  =  n{a  -  b){b  -  c){c  r-  a), 
and  if  a  =  2,  6  =  1,  c  =  0,  this  reduces  to 
2  =  -  2  •  n, 
whence  w  =  —  1. 

6.    .-.  the  expression  equals  —  {a  -  b)  {b  —  c)  {c  —  a). 

Check  by  letting  a  =  3,  6  =  2,  c  =  l,  or  other  values. 

EXERCISES.    CLXXV. 

Factor  the  following : 

1.  ^x\y-z).  2.  2a'(6'-c2). 

3.  ^x\7/-z).  4.  ^a\b^-c^). 

5.  (^af  -  :^a^  6.  ^a"^  -  2  %a%\ 

7.  {%a)  {%ab)  —  abc.  8.  %ah  {a  -\- b) -{- 2  abc. 

9.  S«  (J)''  +  c2)  +  2  abc.  10.  %a  (b  -  c)'^  +  8  abc. 

11.  %a{b-\-cY-^abc.  12.  a^  -  b^  j^  c^  +  ^  abc. 

13.  ^{a-b){a  +  b  -  cf.  14.  %(a  -  b){a -\-b -2  o)^. 


388  ELEMENTS   OF   ALGEBRA. 

15.    %a^(b  +  c)  +  abG'S,a.  16.    4:  a^""  -  (a""  +  b""  -  c^. 

17.  (^xy  +  ^x^  -  ^(x +  yy. 

18.  (S,xy  +  %x' -  ^(x  +  yy. 

19.  a^ -\- b^ -\- c^  —  3  abc.    One  factor  must  be  a  ±  5  or  ^a. 
(Why  ?) 

20.  %a^  -^  3(a  -\-  b){b  -\-  c){c  +  a). 

21.  %(a  —  by,  "Z  referring  to  a,  b,  c. 

22.  ^(a^  -  by,  2  referring  to  a,  b,  c. 

23.  {%a){-^ab)-{a  +  b){b  +  c){c  +  a). 

24.  X  {f  -  z^)  +  7/  (z^  -x^)-j-z  (x^  -  if). 

25.  {s  -  ay  +{s-  by  -  (.s  -  cy,  where  s  =  a  +  b  •}- c. 

26.  2a\b  -  c),  i.e.,  a'ib  -  c)  +  ^'^(c  _  a)  +  c'^{a  -  b). 

27.  {x-ay{b  -c)  +  {x-by{c-a)  +  {x-cy{a-b). 

28.  {a  +  b){a-  by  ^  (b  +  c)  {b  -  cy  +  ((;  +  a)  (c  -  ay. 

29.  2  (a  -  Z*)  (a2  +  Z*^)^  ^-.g.^  (^^  _  ^^  (^^2  _^  ^2^ 

+  {b-c)  (b^  +  c2)  +  (c  -  «)  (c^  +  6^2). 

30.  (x4-2/  +  ^)'-(^  +  2/-^)'-(^  +  -^-^)' 

-(z  +  x-  tjy. 

31.  (ic  -  a)  (x  -b)(a-  b)  +  (x  -  b)  {x  -  c)  (b  -  c) 

+  (x  —  c)(x  —  a)  (c  —  a). 

32.  a(b  +  c)  (b^  +  c''-a^)-\-b(c-^  a)  (c^  +  a^  -  b^) 

+  c(a  +  b)(a^-^b^-c''). 

33.  Find  three  factors,  only,  of  (x  —  yy"  +  ''  f  (y  —  zy''  +  ^ 

+  {Z  —  CC)2»  +  1. 

34.  Also  of  (5ic)2"  +  i  -  :§a;2"  +  i,  2  referring  to  x,  y,  z. 


APPENDIX.  389 

The  type  Sx^  +  S  2  xy,  the  square  of  a  polynomial. 
Since  C^xf  =  %x''  +  '^2xy  (p.  385),  it  follows  that  ex- 
pressions in  the  form  of  "^x^  -\-^2xy  can  be  factored. 

E.g.,      x2  4-  2/2  +  22  +  2 x?/  +  2 ?/2  +  2  zx  =  (X  +  ?/  4-  z^. 

Check.     9  =  82. 

Similarly,  4  a2  +  9  62  +  c2  -  12  a6  -  6  6c  +  4  ca  =  {2  a  -  3  6  +  c)2.   , 

Check.     Let  a  =  6  =  1,  c  =  2.     Then  1  =  12. 

EXERCISES.    CLXXVI. 

Factor  the  following : 

1.  4  x^  +  9  2/2  +  1  +  12  ic?/  +  6  //  +  4  ic. 

2.  1  +  4  «.2  +  9  6*  +  4  «;  +  6  ^.2  +  12  ah\ 

3.  4  +  16  a^  +  25  Z>*  +  16  a  +  20  U'  +  40  ah\ 

4.  5a;2  +  2/4  +  9  +  2.T?/2V5  +  6ic  V5  +  62/'. 

5.  a-^  +  //  +  9  c«  +  ^'  +  2  a^  +  6  ac^  +  2  a^^  _^  g  ^,2^3 

+  2  ^2^  +  6  c^f^. 

The  following  miscellaneous  exercises  review  some  of  the 
elementary  cases  of  factoring. 

6.  4  x'^  +  8  iy  +  9  ?/^  7.    4  x^  -  4  a^2^2  _^  9  y4_ 
8.    5  0^2  _^  5  +  3  ic  +  3  ic^  9.    xhf  -  4  .t'-^  +  4  -  y-. 

10.    a^j^h^  -1^  +  2  a^lP-.         11.    1  +  4  a;y  -  4  /  -  xl 
12.    x'^a  +  a?/2  +  hy''"  +  ^ic^.       13.    x^  +  144  —  16  x^  —  9  x\ 

14.  %  4-  2  Z-x  +  2  a?/  +  4  «x.  ■ 

15.  x2  4-  wxy  —  4  i^?/2  —  4  x?/. 

16.  x^  4-  10  X  +  2/'  +  10  ?/  4-  25  4-  2  xy. 

17.  x2  -  12  X  4-  4  y/2  4-  36  -  24  y  +  4  x?/. 


390 


ELEMENTS  OF  ALGEBRA. 


V.  GENERAL  LAWS  GOVERNING  THE  SOLUTION  OF 
EQUATIONS  (p.  152). 

Theorem.  If  the  same  quantity  is  added  to  or  subtracted 
from  the  two  members  of  an  equation,  the  result  is  a7i  equiva- 
lent equation. 

Given         A  =  B,  an  equation,  and  C  any  quantity. 

To  prove    that  A±  C  =  B  ±  C  i?,  2a\  equivalent  equation. 

Proof.  1.  If  for  certain  values  of  the  unknown  quantities, 
A  and  B  take  numerically  equal  values,  it  is 
evident  that  A±  C  and  B  ±  C  must  also  take 
equal  values. 

2.  .*.  any  root  oi  A  =  B  is  also  a  root  of  ^  ±  C 

=  B±a 

3.  If  for  certain  values  of  the  unknown  quantities 
A±C  and  B±C  take  numerically  equal  values, 
it  is  evident  that  A  and  B  must  also  take  equal 
values,  because  we  obtain  their  values  by  sub- 
tracting from  the  equal  values  oi  A±  C  and 
B  dt  C  the  same  number. 

4.  .•.  any  root  of  A  ±  C  =  B  ±  C  is  also  a  root  of 
A  =  B. 

5.  Since  a,ny  root  oi  A  =  B  is  also  a  root  of  ^  ±  C 
=  B±C,  and  any  root  of  A±C  =  B±C  is 
also  a  root  of  ^  =  ^,  it  follows  that  the  two 
equations  are  equivalent. 


Corollary.  Every  equation  can  be  put  into  the  form 
A  =  0. 

For  iu  subtracting  from  the  two  members  of  an  equation  a  quantity 
equal  to  its  second  member,  an  equivalent  equation  is  obtained  of  which 
the  second  member  is  0. 


APPENDIX.  391 

Theorem.  If  the  tivo  members  of  an  equation  are  multi- 
plied or  divided  by  the  same  quantity,  which  is  neither  zero 
nor  capable  of  becoming  zero  or  infinitely  great,  the  result  is 
an  equivalent  equation. 

Given         the  equation  A  =  B,  and  the  factor  C,  which  by 
the  conditions  cannot  be  0  or  infinitely  great. 

To  prove    that  AC  =  BC  is  an   equation   equivalent   to 
A^B. 

Proof.    l.\-A=B,         .'.A-B  =  0. 

2.  .-.  C(A-B)  =  0.  Ax.  7 

3.  Every  root  of  1,  making  A  —  B  =  0,  must  also 
make  C{A  —  B)  =  0,  because  C  is  not  infinitely 
large. 

If  C  =  00,  then  C  {A  —  B)  would  be  undetermined,  by  §  172. 

4.  .'.  every  root  of  1  is  a  root  of  2. 

5.  Conversely,  every  root  of  2,  making  C(A  —  B) 
=  0,  must  also  make  A  —  B  =  0,  because  C  is 
not  zero. 

If  C  =  0,  then  A  —  B  would  equal  §,  an  undetermined  quantity 
by  §  168. 

6.  .".  every  root  of  2  is  a  root  of  1. 

7.  From  4  and  6,  the  equations  are  equivalent. 

The  necessity  for  the  limitations  on  the  value  of  the  multiplier  is 
evident  from  a  simple  example.     In  the  equation 

X2  +  X  =  0 

we  cannot  expect  to  get  an  equivalent  equation  by  dividing  by  x 
or  multiplying  by  -,  for  x  =  0  and  -  =  co,  and  the  simple  equation 

x  +  1  =0 
is  evidently  not  equivalent  to  the  quadratic  equation  x^  +  x  =  0. 


392  ELEMENTS  OF  ALGEBRA. 

Theorem.  If  the  two  members  of  a  rational  fractional 
equation  are  multiplied  by  the  lowest  common  denominator 
of  the  fractions,  the  result  is,  in  general,  an  equivalent 
equation. 

Proof.    1.  The  equation  can  be  transformed  so  that  the 
second  member  is  0. 

2.  The  first  member,  being  a  rational  fractional 

expression,  can  then  be  reduced  to  the  form  --? 

in  which  B  is  the  lowest  common  denominator 
of  the  fractions,  after  they  are  added  and 
reduced,  and  hence  is  prime  to  A. 

A 

3.  .'.  the  equation  can  be  reduced  to  —  =  0,  the 

B 

members  of  which  it  is  proposed  to  multiply 
hy  B. 

4.  There  can  be  no  values  of  th^  unknown  quan- 
tity which  make  A  and  B  zero  at  the  same 
time,  since  B  is  prime  to  A. 

A 

5.  .'.in  order  that  —  =  0  it  is  necessary  and  suffi- 
cient  that  A  =  0.     .'.  the   equation   ^  =  0  is 

A 
equivalent  to  the  equation  —  =  0. 

To  illustrate  the  theorem,  consider  the  following  cases : 

4 
In  the  equation  -  =  x,  it  is  legitimate  to  multiply  by  x,  giving  4  =  x-, 

whence  x  =  -|-  2,  or  —  2,  either  root  satisfying  the  original  equation. 

But  in  the  equation  —  =  1 ,  it  is  not  legitimate  to  multiply  by  x,  for 
then  x2  =  X,  and  x^  —  x  =  0,  whence  x  (x  -  1)  =  0,  x  =  0,  or  1.  But 
X  =  0  does  not  satisfy  the  original  equation,  because  g  does  not  neces- 
sarily equal  1. 

x"^  -}-  5x  -j-  6 
Similarly  we  cannot  solve =  3  by  multiplying  by  x  -|-  2. 

X  -|-  2 


APPENDIX.  393 

Theorem.  If  both  members  of  an  equation  are  raised  to  any 
integral  power,  the  resulting  equation  contains  all  of  the 
roots  of  the  given  equation,  but  in  general  is  not  equivalent 
to  it. 

Given         the  equation  A  =  B. 

To  prove  that  the  equation  A"^  =  B""  contains  all  of  the 
roots  of  the  equation  A  =  B,  but  in  general  is 
not  equivalent  to  it. 

Proof.    1.  From  ^  =  ^  it  follows  that  A  —  B  =  0. 

2.  From  ^'"  =  ^"  it  follows  that  ^'"  -  B""  =  0. 

3.  But  whether  m  is  odd  or  even,  A'"  —  B""  con- 
tains the  factor  A  —  B. 

4.  .".  equation  2  becomes 

{A  -  B)  (^'"-1  +  A"'-''B  +  ...)=  0, 
and  is  satisfied  hy  A  =  B. 

5.  .".  equation  2  contains  the  roots  of  equation  1. 

6.  But  from  equation  4, 

and  hence  A"^  =  B'"  contains  other  roots  than 
A  =  B,  and  hence  is  not  equivalent  to  it. 

To  illustrate  let  x  =  2  ; 

squaring,  x'^  =  4, 

an  equation  containing  the  root 

x  =  2, 

but  also  the  extraneous  root 

x=  -2. 

If  we  cube,  x^  =  8, 

and  this  again  contains  the  root 

x  =  2, 

but  it  also  contains  the  extraneous  roots 

X  =  -  1  ±  V  ~3. 


394  ELEMENTS  OF  ALGEBRA. 

VI.  EQUIVALENT  SYSTEMS  OF  EQUATIONS  (p.  185). 

It  has  been  shown  that  the  solution  of  a  system  of  equa- 
tions is  made  to  depend  upon  the  solution  of  a  second  sys- 
tem derived  from  the  first.  But  it  has  not  yet  been  shown 
that  extraneous  roots  are  not  introduced  by  this  operation. 

Two  systems  of  equations,  each  having  the  same  roots  as 
the  other,  are  called  equivalent  systems. 

Theorem.      Given  a  system  of  two  equations 

(1)  f(x,y)=0,     F(x,y)  =  0, 
and  a,  b  two  numbers  (b  ^  0),  then 

(2)  a.f(x,y)  +  b-r(x,y)  =  0,     f(x,y)  =  0 

is  an  equivalent  system. 

Proof.  1.  •.'  a  solution  of  system  (1)  makes  both  f(x,  y) 
and  F(x,  y)  equal  zero,  it  makes  both  a  -/(x,  y) 
and  b  ■  F(x,  y)  equal  zero,  and  hence  satisfies 
system  (2). 

2.  •.•a  solution  of  system  (2)  makes 

and  a.f{x,y)^b-F{x,y)^^, 

it  must  therefore  make  a  -/(x,  y)  =  0, 

and  hence,  b-  F(x,  y)  =  0, 

and  hence,  F(x,  y)=0,  ■,•  b^O. 

Hence,  it  is  a  solution  of  system  (1). 

This  theorem  justifies  the  solution  of  two  simultaneous 
linear  equations  by  addition,  subtraction,  and  substitution. 
For  it  shows  that  we  may  multiply  the  members  by  any 
numbers  (a,  b),  add  or  subtract  (since  b  may  be  negative) 
the  equations  member  for  member,  and  combine  this  result 
with  the  equation  f(x,  y)  =  0. 


APPENDIX. 


395 


VII.  DETERMINANTS  (p.  198). 

The  practical  solution  of  simultaneous  linear  equations, 
while  possible  by  the  methods  already  given,  is  frequently 
tedious.  For  this  reason  mathematicians  often  resort  to  a 
simpler  method,  that  of  determinants. 

The  theory  of  determinants  is  comparatively  modern,  and 
although  it  is  not  practicable  to  enter  into  the  subject  at 
any  length  at  this  time,  the  elementary  notions  are  so 
simple  and  so  helpful,  and  the  applications  so  common, 
that  a  brief  presentation  of  the  subject  will  be  of  value. 


The  symbol 


a^bo 


is    merely   another  way   of  writing 


aib^  —  a^bi.     The  symbol  is  called  a  .determinant,  and  the 
letters  a^,  a^,  b^,  b^  are  called  its  elements. 

This  is  a  determinant  of  the  second  order ;  i.e.,  there  are  two  ele- 
ments on  each  side  of  the  square.  It  will  be  noticed  that  the  expanded 
form  is  simply  the  difference  of  the  diagonals. 

In  a  determinant,  the  horizontal  lines  of  elements  are 
called  roivs,  the  vertical  ones  columns. 

In  the  above  determinant  the  rows  are  ai,  h\  and  a^,  62  ;  the  columns 
are        and  ,  . 

&2 


When   the   determinant 


a^b^ 


is  written    in   the    form 


a^^  —  a^bi  it  is  said  to  be  expanded. 

It  is  understood  that  the  expanded  form  is  to  be  simplified  in  all 

12  31 
cases.      E.g.,  while  =2-7  —  5-3,  the  result  should  be  stated 

as  —  1.  ' 


EXERCISES.    CLXXVII. 

1.    Expand  the  following  determinants  : 


a  b 
y  X 


X  y 
b  a 


ay 
b  X 


X  b\ 

y  a| 


396 


ELEMENTS   OF  ALGEBRA. 


2.    Also  the  following  : 


13 
42 

) 

1  4 
32 

> 

23 
41 

) 

24 
31 

3. 

Also  the  following : 

10 

50 

0    0 

0  19 

Oa 

00 

20 

'         7  0 

> 

18  30 

) 

0  27 

J 

0  h 

) 

a  h 

4.  From  exs.  1  and  2,  state  what  changes  can  be  made 
in  a  determinant  of  the  second  order  without  changing  its 
value. 

5.  From  ex.  3,  what  is  the  value  of  a  determinant  if 
either  a  row  or  a  column  is  made  up  of  zeros  ? 

6.  Expand  the  following  determinants  : 


12  6 

57 


8  1 
23 


56 

78 


7.   Expand  the  following  determinants 


«i  ^1 


a^  a^ 
h  h 


8.  From  ex.  7,  state  the  effect  on  the  value  of  a  deter- 
minant of  the  second  order  of  changing  the  rows  into 
columns  and  the  columns  into  rows. 

9.  Expand  tlie  following  determinants  : 


ai  b^ 

«!  +  h 

h. 

«!  -b^ 

h 

^2   f>2 

> 

a^  +  b^ 

h. 

5 

a^  —  ^2 

h. 

10.  From  ex.  9,  state  the  effect  on  the  value  of  a  deter- 
minant of  the  second  order,  of  increasing  the  elements  of 
one  column  by  the  corresponding  elements  of  another,  or  of 
diminishing  the  elements  of  one  column  by  the  correspond- 
ing elements  of  another. 


I 


APPENDIX. 


397 


11.    Expand  the  following  determinants 


aiZ>i 

2a^b^ 

mai  bi 

as  h 

? 

2a^h^ 

} 

ma^  b^ 

12.  From  ex.  11,  state  the  effect  on  the  value  of  a  deter- 
minant of  the  second  order,  of  multiplying  the  elements  of 
a  column  by  any  number. 

A  determinant  of  the  second  order  is  no  more  easily 
written  than  is  its  expanded  form.  But  one  of  the  third 
order  (one  with  three  elements  on  the  side  of  the  square) 
is  materially  more  condensed  than  is  its  expanded  form. 

<Xi  Z*!   Ci 

is   the  general  form  of  a  deter- 


The   symbol 


^2  ^2   ^2 

(^z  bg  C3 
minant  of  the  third  order,  and  it  stands  for 


<^J^2^3  +  fh^'zCi  +  ^z^xC^  —  a^b^fix  —  a^b^c^  —  a-fi^c^. 


The   expansion  of  a  third  order  determinant  is  easily 

written   by    following    the  ^ 

arrows  in  this  arrangement.  ^-""^  ^^x 

This  method   of  expansion 

holds    only  for   determinants 

of  the  second  and  third  orders, 

all  that  we  shall  treat  in  this 

work. 

The  fact  that  the  student  is 
rarely  called  upon,  in  elementary- 
algebra,  to  solve  a  system  of  more 
than  three  simultaneous  linear 
equations  makes  it  undesirable  to  enter,  at  this  time,  upon  the  theory 
of  determinants  of  an  order  hii-her  than  the  third. 


398 


ELEMENTS   OF  ALGEBRA. 


EXERCISES.    CLXXVIII. 

1.    Expand  the  following  detefrminants,  the  rows  of  the 
first  being  the  columns  of  the  second : 


123 

147 

456 

J 

2  58 

789 

3  69 

2.    Also  the  following  : 


«!   hy  Ci 

ai  (^  ag 

a^  h^  Cg 

J 

h  b,  b. 

as  bs  Cg 

Ci      C2     Cg 

3.  From  exs.  1  and  2,  state  the  effect  on  the  value  of  a 
determinant  of  the  third  order,  of  changing  the  rows  into 
columns  and  the  columns  into  rows. 

4.  Expand  the  following  determinants  : 


246 
135 

7  89 

J 

2  +  446 
1+335 

7  +  889 

' 

646 

435 

15  8  9 

5.    Also  the  following  : 


^1  &1  Ci 

^2  b^  C2 

J 

as  ^3  C3 

(Xi  +  b^       bi 

«2  +  ^2  ^2 

as  +  ^3         ^3 


I 


6.  From  exs.  4,  5,  state  the  effect  on  the  value  of  a  de- 
terminant of  the  third  order,  of  increasing  the  elements  of 
one  column  by  the  corresponding  elements  of  another. 

7.  Expand  the  following  determinants  : 


0  ab 

000 

0  cd 

> 

a  b  c 

0  ef 

def 

APPENDIX. 


399 


8.  From  ex.  7,  what  is  the  value  of  a  determinant  of  the 
third  order  if  either  a  row  or  a  column  is  made  up  entirely 
of  zeros  ? 

9.  Expand  the  following  determinants  : 


«!  bi  Cx 

mai  bx  Ci 

^2  h  C2 

) 

ma.i  ^2  <^2 

dz    h   Cg 

mag  b^  Cg 

10,  From  ex.  9,  state  the  effect  on  th^  value  of  a  deter- 
minant of  the  third  order  of  multiplying  the  elements  of  a 
column  by  any  number. 

In  the  preceding  exercises  certain  general  theorems  have 
been  proved  by  the  student  for  determinants  of  the  second 
and  third  orders.  These  will  now  be  presented  formally, 
the  proof,  however,  referring  only  to  determinants  of  these 
orders. 


Theorem.     The  value  of  a  determinant  is  unchanged  if  the 
rows  are  changed  to  columns  and  vice  versa. 


Given       the  determinant 


Proof.       Each  expands  into 

dib^Cz  +  a^b^Cx  +  «3^i<?2  —  «^3^2Ci  —  a^biC^  —  axb^c^. 

The  proof  for  the  determinant  of  the  second  order  is  left 
for  the  student.     Take  the  determinants 


ax  bx  Cx 

a^  ^2  C2 

. 

«3  h  Cz 

ax  a^  a^ 

determinant 

bx  b^  bs 

Cx      C2      Cg 

ax  bx 

^2  ^2 


and 


ax  a^ 
bx  62 


and  expand. 


400 


ELEMENTS  OF  ALGEBRA. 


Theorem.  If  each  element  of  a  column  {or  row)  of  a 
determinant  is  multiplied  by  any  factor,  the  determinant 
is  multiplied  by  that  factor. 

Proof.    Consider  the  determinants 


ai  bj_  Ci 

ai^i 

^2  h  C2 

and 

«3  ^8  C3 

a^b^ 

By  the  law  of  expansion  every  term  of  the  ex- 
panded form  contains  one  a  (i.e.,  a^,  a^,  or  a^) 
and  only  one ;  hence,  if  every  a  is  multiplied  by 
m,,  the  7n  will  appear  once  and  only  once  as  a 
factor  of  every  term  of  the  expanded  form. 
Similarly,  for  any  column  or  row. 


13  2 
For  example,  consider  the  determinant  L  „ 

6  / 


If  we  multiply  either 
This 


column  or  either  row  by  2,  the  determinant  is  multiplied  by  2 
is  seen  by  expanding 


62 

3    4 

64 

3    2 

10  7  ' 

5  14  ' 

57  ' 

10  14 

the  results  being  42  —  20  =  22  in  each  case,  while  the  original  deter- 
minant equals  21  —  10  =  11. 

Theorem.     If  a  column  (or  row)  is  m,ade  up  entirely  of 
zeros,  the  determinant  equals  zero. 

Proof.  As  in  the  precefding  theorem,  every  term  of  the 
expanded  form  contains  an  a ;  hence,  if  every  a 
is  zero,  the  expanded  form  vanishes.  Similarly, 
for  any  other  row  or  column. 

1  20 
This  is  seen,  for  a  special  case,  in  the  determinant    4  7  0    which 

360 
expands  into  l-7-0-t-4.6.0-f3.0-2-l-6-0-4-2-0-3-7-0  =  0. 
The  same  may  be  seen  in  the  case  of  determinants  like 
1241         103 


00 


02  ' 


etc. 


APPENDIX. 


401 


Theorem.  If  each  element  of  a  column  is  multiplied  by 
any  number,  and  a^ded  to  the  corresponding  element  of  any 
other  column,  the  value  of  the  determinant  is  not  changed. 

«-!  bi  Ci 

a<i  c>2  C2 
^3  b^  Cs 


«1 

+  mbi 

h 

c. 

^2 

+  mbz 

h. 

C2 

as 

4-  mbs 

h 

C3 

Given        the  determinant 


To  prove  that  it  equals  the  determinant 

Proof.       Expanding  the  second  determinant,  it  equals 

(^i  +  mb^  b^c^  +  (0-2  +  ^^^2)  ^3^1  +  (^3  +  mh^  b^c^ 
—  (a^  4-  mb^  b^Px  —  {a^  +  mh^  b^c^  —  {ax  +  mb^  b^c^ 
which  equals 

ai^2^3  +  «2^3'?l  +  ^3^lC2  —  ^3^1  —  «2^lC3  "  «1^3C2 

the  other  terms  all  cancelling  out. 

That  is,  the  two  determinants  are  equal. 

The  proof  is  the  same  whatever  columns  (or 
rows)  are  taken,  and  for  the  second  order  as  well 
as  for  the  third. 

Corollaries.  1.  The  elements  of  any  coluinn  (or  row) 
may  be  added  to  or  subtracted  from  the  corresponding  ele- 
ments of  any  other  column  (or  r^w)  without  changing  the 
value  of  the  determinant. 

For  m  may  equal  1  or  —  1. 

2.  If  ttvo  columns  (or  rows)  are  identical,  the  determinant 
equals  zero. 

For,  if  the  elements  of  one  are  subtracted  from  the  corresponding 
elements  of  the  other,  a  column  (or  row)  will  be  composed  of  zeros. 

3.  If  the  elements  of  one  column  are  the  same  inultiples 
of  the  corresponding  elements  of  another,  the  deternninant 
equals  zero.     (  Why  ?) 


402 


ELEMENTS   OF   ALGEBRA. 


Dlustrative  problems.     1.  Expand  the  determinant 

27  25 
42  41 

Subtracting  the  second  column  from  the  first,    the  determinant 

1 2  25 1 
equals   L  .  J  =  82  -  25  =  57. 

1 1  41 1 

This  is  much  easier  than  finding  the  value  of  27  •  41  —  42  •  25. 


2.  Expand 


8  21 
6  15 


Factoring  the  second  column  by  3,  and  then  subtracting  it  from 
the  first,  we  have 


65 


''  =  3|!  ^1  =  3(5-7) 


1  5 


3.  Expand 


10  17  3 
20  16  4 
30  15  5 


Subtracting  the  first  row  from  the  second  and  that  from  the  third, 


10  17  3 
10  -1  1 
10  -1  1 


=  0. 


(Why  ?) 


General  directions  for  exj)anding  determinants. 

1.  Remove  factors  from  columns  or  rows. 

2.  Endeavor  to  make  the  absolute  values  of  the  elements 
as  small  as  possible  by  subtracting  corresponding  elements  of 
rows  or  columns,  or  multiples  of  those  elements. 

3.  Endeavor  to  bring  in  as  inany  zeros  as  possible. 

4.  Endeavor  to  make  the  elements  of  two  columns  (or 
rows)  identical,  so  that  the  determinant  may  be  seen  to  be 
zero  (if  that  is  its  value)  without  expanding. 

5.  After  thus  siinplifying  as  much  as  possible,  expand. 


APPENDIX. 


403 


EXERCISES.    CLXXIX. 

Expand    the   determinants    or   prove    the    identities    as 
indicated. 


1. 

121 
112 
21  1 

2. 

97  96 
63  62 

. 

3. 

3    9 
13  39 

• 

Oil 

14    1 

aba 

4. 

125 
137 

5. 

4  16    4 
97     5  17 

6. 

a2  b^  ab 
a^  b'  ab^ 

• 

1  a  a^ 

7. 

1  b  b^ 

1  c  c' 

=  {a-b)ib-  c)  (c  -  a). 

a  0  c 

a  -\-  b      c          G 

8. 

a  b  0 
0  b  c 

=  2  abc.                 9. 

a      b  -\-  e      a 
b          b      c  -\-  a 

=  4  abc. 

p  +  c2       ab           ca 

10. 

ab      c^  +  a^       be 
ca          be       a^  +  b'^ 

=  Aa%^c^ 

Application  of  determinants  to  the  solution  of  a  system  of 

two  linear  equations. 

On  solving  the  system 

a^x  +  b^y  =  Ci, 

a2X  +  b2y  =  C2, 

the  roots  are  found  to  be 

11  = 

«] 

Gi  - 

2_ 

a^ 

^. 

aib2  —  ^2^1 


404  ELEMENTS  OF  ALGEBRA. 

It  is  at  once  seen  that 

,  made  up 


aibi 
a2  02 


64 

11 

-t 

11 

16 

-7 

—  7 

3 

11 

14 

11 

17 

-  7 

10 

-7 

1.  Each  denominator  is  the  determinant 
of  the  coefficients  ofx  and  j. 

2.  The  numerator  for  x  is  the  same  determinant  ivith  c 
put  for  a  (the  coefficient  of  x). 

3.  The  numerator  for  j  is  also  the  same  determinant, 
ivith  c  put  for  b  (the  coefficient  of  y). 

Illustrative  problem.     Solve  the  system 
3a;  +  lly/  =  64, 
nx-ly  =  U. 


Herex^'^^-^U      '^    -  ^  L  16(- 28  -  11)  ^  8,^^  ^ 

2(-49-55)  -104 

.-.  y  =  5,  by  substitution. 

EXERCISES.    CLXXX. 

Solve  by  determinants,  checking  in  the  usual  way. 

.     24^  93       .  „    49      56      ^ 

1. \ =  41  2. =  7. 

X         y  ^  X         y 

1?-31  =  1.  ^-^  =  21i. 

X         y  X         y  ^ 

3.    23£c-302/  =  2.  4.    23a^  +  10  2/  =  252. 

10  a:;  +  7  2/  =  61.  19  ;z;  +  17  y  =  154.7. 

5.    41x-'dly  =  4..  6.    235  £c- 234?/:.=  236. 

43  a;  +  39  2/  =  82.  411  x  +  410  ?/  -.  412. 

7.    52  a; -39  2/ =  13.  8.    0.5  ic  -  0.3  ?/- 0.021. 

18  a^  +  18  3/  =  15.  0.6  X  +  2  y  ^  0.332. 


APPENDIX. 


405 


Three  linear  equations  with  three  unknown  quantities. 
On  solving  the  system 

a^x  +  biT/  +  Ciz  =  di, 
a^x  4-  b0  +  02Z  =  c?2> 
a^x  +  b.^//  +  c^z  =  ds, 
the  roots  are  found  to  be 

d^b^Cs  +  d^br^Ci  4-  dsbjC^  —  dJ)2C^  —  djj^c^  —  d^^c^ 

^i<^2<^3  +  «'2fi?3Ci  +  a^diCo,  —  a^d^Ci  —  a^d^c^  —  a-^d^c^ 


X  = 


(^\hlCz  +  «2^3Cl  4-  «3^lC2  —  «3^2Cl  —  ^^a^l^S 

^i^2f^3  +  ciibzdx  +  a^bidi^  —  a^b^di  —  a.^xdz 


-  a-Jy^d^ 


dih 

Cl 

d,M 

C2 

chbs 

C3 

aih 

^1 

a2  62 

C2 

a3&3 

C3 

ai 

di 

Cl 

a2 

d. 

C2 

as 

ds 

C3 

tti 

61 

Cl 

ag 

62 

C2 

a3 

63 

C3 

ai 

W 

dl 

a2 

&2 

d2 

ag 

&3 

ds 

ai 

&1 

Cl 

az 

62 

C2 

as 

&3 

C3 

ai^a^s  +  a^bsCi  +  aaZfiCg  —  asb^c^^  —  ag^i^s  —  «i^3C2 

It  is  at  once  seen  that  the  same  law  already  set  forth 
holds  here,  and  that  the  roots  may  be  expressed  thus  : 


y  = 


It  is  thus  seen  that  the  roots  of  three  linear  simultaneous 
equations  can  be  written  down,  in  the  determinant  form,  at 
sight.  It  then  becomes  merely  a  matter  of  simplifying. 
Whether  it  is  easier  to  solve  by  determinants,  or  to  solve 
by  elimination  through  addition  and  subtraction,  depends 
largely  on  the  size  of  the  coefficients.  If  the  coefficients 
are  small,  there  is  usually  no  advantage  in  using  deter- 
minants ;  if  they  are  large  there  is  often  a  great  gain.  In 
the  problems  on  the  next  two  pages  the  coefficients  are  not 
in  general  large  enough  to  make  it  worth  while  to  use 
determinants  except  for  practice. 


406 


ELEMENTS   OF   ALGEBRA. 


Illustrative  problem.     Solve  the  system 

13x  +  lly  -\-llz  =  0, 

17  a? +  15  2/ +  80;^  =  30. 
The  common  denominator  for  ic,  ?/,  z  is 


11  9  33 

2  9  33 

2  9  33 

19  33 

13  11  71  = 

2  11  71 

= 

0  2  38 

=  2-2 

0  1  19 

17  15  80 

2  15  80 

04  9 

04  9 

=  2  .  2  (9  -  4  •  19)  =  -  4  ■  67. 
(How  is  the  second  determinant  obtained  from  the  first  ?  the  third 
from  the  second  ?  and  so  on.) 

The  numerator  for  x  is 


52  9  33 

26  9  33 

Oil  71 

=  2-5 

0  11  71 

=  10 

20  15  80 

3  3  16 

17    7! 

11  71 

0  13 


10(26-11  -13 

-  10  •  134. 

-10    134      ^ 
=  5 

-4-67 


17-71  -3.11-7) 


11  52  33 

13  0  71 

=  2 

17  30  80 

The  numerator  for  y  is 

11  26        0 

13    0      32    =  2(17-32-26-29.  13-26-11 -15-32) 

17  15  -29 

=  -  2    938. 
.       _  -^-^ 

"We  may  now  find  z  by  substitution.     Or  the  numerator  for 


11  9  52 

2  9  26 

0  -2  26 

13  11  0 

=  2 

2  11  0 

=  2-2 

1  11  0 

17  15  30 

2  15  15 

0   4  15 

by  factoring  by  2  and  subtracting  the  second  row  from  each  of  the 

others.     This  equals  4  .  134. 

4  •  1.34 

.-.  z  = =  -  2. 

-  4  -  67 


APPENDIX.  407 

EXERCISES.    CLXXXI. 

Solve  by  determinants,  checking  all  numerical  results  in 
the  usual  way. 

2.    5ic-3^  =  3. 

2x  +  y  =  5. 
3  2/  +  «  =  7.5. 

X       .       y  rr     ^ 

3  +  2-^=^-^- 

2      3^4 

6.    7x-3ij-2z  =  lQ. 
2x-5  7j-{-Sz  =  S9. 

5x  +  1/  -{-  5z  =  31. 

8.   3ic  +  3  2/ 4- 3  2;  =  144. 

9x  -\-y  —  z  =  154. 

10.    a^^  +  Z^'^^z+c^^^ft  +  Z'  +  c. 
aa:  +  6?/  +  cs;  =  1. 

X  +  y  +  z  =  0. 

12.    2a;-32/  +  42;  =  -18. 
3a;  +  4?/-5^  =  34. 

X  -{-y  +  z  =  0. 

13.    123x  +  17  2/-139;s  =  l. 

51  ic  +  37  2/ -  97  ^  =  -  9. 
5  a^  +  31 2/ -  35  s  =  1. 


1. 

x  +  y  =  10. 

y-{-z  =  10. 

x-\-  z  =  6. 

3. 

5  +  f  =  3. 
a       0 

0        z 

5  +  5  =  4. 
a       c 

5. 

12x4-7  7/ =  109. 

52/ -2.^  =  11. 

4a; +  3;^  =  26. 

7. 

4:X-{-9y-\-z  =  16. 

2a; +  32/ +  .^  =  4. 

a;  +  2/  +  ^  =  l- 

9. 

^'■^o;  +  q^y  +  r^t;  =  s\ 

p^x  +  </^2/  +  ^'^^  —  "^^• 

^ic  +  2'?/  +  r;s  =  s  . 

11. 

3cc  +  42/  +  2;^  =  47. 

5x-3y  +  7z  =  4:l. 

7x-2y-5z  =  24.. 

Pa- 


408  ELEMENTS   OF   ALGEBRA. 

VIII»     GRAPHIC   REPRESENTATION  OF  LINEAR 
EQUATIONS  (p.  202). 

In  the  annexed  figure  the  two  lines  XX^  and  YY\  inter- 
secting at  right  angles  at  0,  are  called  rectangular  axes. 

A  segment,  OA,  on  OX  is  called 
the  abscissa  of  any  point,  as  Pi, 
on  a  perpendicular  to  XX'  at  A. 
A  segment,  OB,  on  0  Z  is  called 
the  ordinate  of  any  point,  as  Pj, 
X    on  a  perpendicular  to  FF'  at  P. 
The  abscissa  and  ordinate  to- 
gether are  called  the  coordinates 
of  the  point,  the  abscissa  always 
being  named  first. 
Abscissas  to  the  right  of  0  are  called  positive,  those  to 
the  left  negative.     Ordinates  above  0  are  called  positive, 
those  below  negative. 

E.g.^  in  the  figure  the  coordinates  of  Pi  are  3,  2 ;  those  of  P^  are 
—  4,  1 ;  those  of  P3  are  —  2,  —  3 ;  those  of  P4  are  2,  —  2  ;  those  of 
A  are  3,  0  ;  those  of  0  are  0,  0.  The  ordinate  of  any  point  on  XX' 
is  evidently  0,  and  the  abscissa  of  any  point  on  YY'  is  0  also. 

Hence,  when  the  axes  are  given  a  point  in  their  plane  is 
fixed  when  its  coordinates  are  known.  Conversely,  when 
a  point  is  fixed  its  coordinates  with  respect  to  any  given 
axes  are  evidently  fixed  also. 

A  point,  as  Pi,  is  designated  by  its  coordinates. 

Thus,  Pi  is  designated  by  (3,  2),  P2  by  (-  4,  1),  P3  by  (-  2,  -  3), 
and  P4  by  (2,  -  2). 

If  the  coordinates  are  unknown,  they  are  designated  by  x  and  ?/, 
the  point  being  designated  by  the  symbol  (x,  y).  That  is,  if  it  is 
desired  to  speak  of  two  general  points,  as  we  speak  of  two  unknown 
quantities  in  algebra,  they  may  be  designated  either  as  Pi,  P2,  or  as 

(«1,  Vl),    (X2,   ?/2). 


.    .       APPENDIX.  409 

EXERCISES.    CLXXXII. 

In  each  exercise  draw  a-  pair  of  rectangular  axes  and 
take  I  inch  as  the  unit  of  measure  for  laying  off  the  coor- 
dinates. 

1.  Represent  the  points  (2,  5),  (—  4,  —  7). 

2.  Also  (5,  0),  (0,  5). 

3.  Also  (0,  0),  (2,  2),  (-  4,  -  4).  Join  these.  Do  they, 
or  do  they  not,  lie  in  the  same  straight  line  ? 

4.  Similarly  for  the  points  (-3,0),  (0,3),  (3,0),  (0,-3). 

5.  What  kind  of  a  figure  is  formed  by  joining,  in  order, 
the  points  (2,  4),  (-  2,  4),  (-  2,  -  4),  (2,  -  4)? 

6.  Also  the  points  (6,  3),  (3,  3),  (3,  -  5)? 

The  graph  of  an  equation.  The  equation  y  =  x  —  1  is 
satisfied  if 

x  = 1,        0,  1,  2,  . . . 

while  7/  = 2,  -1,  0,  1,  •••. 

The  points  (—1,-2),  (0,-1),  (1,  0),  •  •  •  may,  therefore, 
be  thought  of  as  lying  on  a  line  representing  this  equation. 

Hence,  in  the  figure  the  line 
MN  is  considered  the  graphic 
representation  of  the  equatioil 
y  =  x  —  l.  Such  a  graphic  rep- 
resentation is  called  the  graph 


of  the  equation.  ^  ' 

The  word  locus  is  sometimes 
used  instead  of  graph.     Locus  M 

is  a  Latin  word  meaning  jyZace, 
and  the  line  is  the  place  where  the  points  are  found. 
Strictly,  therefore,  it  is  the  graph  of  the  equation  and  the 
locus  of  the  points  which  we  have. 


410  ELEMENTS  OF  ALGEBRA. 

In  a  case  like  that  of  y  =  x  —  1,  y  is  a,  function  of  x. 
Hence,  the  abscissas  represent  the  variable  x,  and  the  ordi- 
nates  represent  the  function. 

A  simple  equation  containing  two  unknown  quantities  can 
always  be  represented  graphically  by  a  straight  line. 

This  is  the  reason  why  it  is  called  a  linear  equation,  a 
term  which  has,  however,  been    extended   to   include  all 
equations  of  the  first  degree. 

Hence,  it  is  necessary  to  locate  only 
/     two  points  to  determine  the  graph  of  a 
simple  equation. 

The  easiest  plan  usually  consists  in 
letting  X  =  0  and  finding   the   corres- 
ponding value  of  y ;    then  letting  y  =  0 
and  finding  the  corresponding  value  of  x. 

E.g.,  to  draw  the  graph  of  the  equation  2x  —  Sy  =7.  Ifx  =  0, 
then  2/  =  —  I ;  if  y  =  0,  then  x  =  |.  Hence,  draw  a  line  through 
(0,  —  I)  and  (I,  0),  as  in  the  figure. 

Since  the  line  represents  the  equation  it  is  evident  that 
the  coordinates  of  any  point  on  the  graph  satisfy  the  equa- 
tion.    That  is, 


a  single  linear  equation  in- 
volving two  unknown  quan-  ^ 
tities  has  an  infinite  number 
of  roots. 


a  single  straight  line  has  an      I 
infinite  number  of  points. 


EXERCISES.    CLXXXIII. 

Draw  the  graphs  of  the  following  equations : 
1.    X  —  y  =  0.  2.    X  +  y  =  0. 

3.    2x-y  =  8.  4.    7x-4:y  =  10. 

5.    -2x-\-3y  =  5.  6.    16x-{-2y  =  7. 


APPENDIX. 


411 


It  is  also  apparent  that  although  a  single 


linear  equation  has  an  in- 
finite number  of  roots,  two 
linear  equations  involving 
two  unknown  quantities  have 
in  general  but  one  common 
pair  of  roots. 

For  example,  the  two  equa- 
tions 

(a)  x-^2y  =  S 

(b)  2x  +  y  =  l 
have  the  graphs  a.  and  b. 

The  two  equations  have 
the  common  pair  of  roots 
whose  values  are 

x  =  2,  2/  =  3. 


straight  line  has  an  infinite 
number  of  points,  two  straight 
lines  have  in  general  but  one 
common  point. 


The  two  graphs  have  the 
common  point  whose  coordi- 
nates are 

x  =  2,  ?/  =  3. 


Hence,  two  linear  equations  involving  two  unknown 
quantities  can  be  solved  by  means  of  graphs,  although 
this  is  not  advisable  in  practice. 


EXERCISES.    CLXXXIV. 


Draw  the  graphs  of  the  following  pairs  of  equations  and 
show  that  the  intersections  represent  the  solutions. 


1.    X  -\-y  =  0. 
X  —  y  =  0. 

3.    3x-^5y  =  12. 
x  +  y  =  2. 


2.    5x-{-2y  =  16. 
3  X  —  y  =  3. 

4.    5x  -\-7y=ll. 

7x-^oy  =  l. 


412 


ELEMENTS   OF   ALGEBRA. 


Discussion  of  Solutions. 


While  in  general 

two  linear  equations  involv- 
ing two  unknown  quantities 
have  a  single  common  pair  of 
roots,  they  may  not,  for  they 
may  be  inconsistent  or  they 
may  be  indeterminate. 

E.g.,  the  equations 

2x  +  Sy  =  6 

2  X  +  3  2/  =  4 
have  evidently  no  common  pair 
of  roots,  since  that  would  make 
6  =  4.      Hence,  they  are  called 
inconsistent. 


Also  the  equations 

?  +  ^  =  l 
2      3 

Sx  +  2y  =  6 

have  no  determinate  solution,  for 

the  members  of  the  second  are 

merely  six  times  those  of  the  first. 

They  are,    therefore,    equivalent 

equations,  and  reduce  to  a  single 

indeterminate  equation. 


two  straight  lines  in  a  iihine 
have  a  single  common  point, 
they  may  not,  for  they  may 
be  parallel  or  they  may  coin- 


E.g.,  the  graphs  of 
2x  +  3y  =  6 
2  X  +  3  2/  =  4 
are  parallel. 


^<^ 


Also  the  graphs  of 

2      3 

3  X  +  2  ?/  =  0 
coincide. 


It  is  a  mistake  quite  often  made  by  students  to  think 
that  it  is  possible  to  solve  anj/  two  equations  like 

8  cc  +  4  ?/  =  5. 
They  may  not  be  simultaneous,  as  in  this  case. 


APPENDIX.  413 


EXERCISES.    CLXXXV. 

Discuss  the  following  systems  of  equations,  solving  if 
possible  and  drawing  graphs  in  all  cases. 

2,     X  -\-  7/  =  4:. 

X  —  y  =  6. 
4.    x-\-3y  =  6. 

X    =   4:. 

6.    6x-\-10y  =  10. 

5      3 

8.    7x  +  35y  =  15. 
3x-^15y  =  S. 

10.    Sx-12y  =  2. 

3      2      12" 

12.    1.02  a; -0.01 2/ =  20.1. 
0.2£c-0.l2/  =  l. 

In  the  same  way  it  may  happen  that  three  equations  in- 
volving three  unknown  quantities  may  be  inconsistent  or 
indeterminate. 

Illustrative  problems.    1.  Solve  the  following  system  : 

1.  9ic  +  62/  +  3^  =  30. 

2.  6£c-f-4^  +  2^  =  20. 

3.  x  +  2y  +  3z  =  14. 

Equations  1  and  2  are  easily  seen  to  be  equivalent.  Hence,  there 
are  only  two  independent  equations,  involving  three  unknown  quan- 
tities, and  they  are  indeterminate. 


1. 

x-\-y  =  l. 

x  —  y  =  1. 

3. 

2/  +  3ic  =  6. 

2/  =  4. 

5. 

2x  +  3y  =  ^. 

2^3        • 

7. 

X- 1.52/  =  10. 

2x-3y  =  B. 

9. 

10  a;  +  6?/ =  5. 

"i  +  y-l. 

3^5      6 

11. 

6  a; +  0.8?/  =  10 

3  a;  +  0.4  2/  =  6. 

414  ELEMENTS   OF  ALGEBRA. 

2.  Solve  the  following  system  : 

1.  6x  +  liy  +  2z  =  Ul 

2.  dx-\-2y  +  Sz  =  22. 

3.  x-\-2y-\-3z  =  14:. 

Equations  1  and  2  are  easily  seen  to  be  equivalent.  Hence,  there 
are  only  two  independent  equations,  involving  three  unknown  quan- 
tities. But  these  two  are  determinate  as  to  x,  for  subtracting  3  from 
2  we  have  8  x  =  8,  .-.  x  =  1.  But  y  and  z  are  indeterminate.  That  is, 
these  two  equations  are  inconsistent  except  for  x  =  1. 

3.  Solve  the  following  system  : 

1.  9x-{-6y-\-3z  =  30. 

2.  6cc  +  4y  +  2^  =  30. 

3.  x-\-2y-\-Sz  =  U. 

Equations  1  and  2  are  easily  seen  to  be  inconsistent ;  for  if  the 
members  of  1  are  multiplied  byf,  Qx  +  4:y  +  2z  =  20  instead  of  30. 

EXERCISES.    CLXXXVI. 

Discuss  the  following  systems  of  equations  with  respect 
to  their  being  indeterminate  or  inconsistent. 

1.    x-^2y-2z  =  0.  2.    5x  +  3  y  -  z  =  -  17. 

2x  —  y  +  z  =  10.  6  .-r  +  4  ?/  =  —  14. 

Sx-{-y-z  =  10.  x  +  y  +  z  =  3. 

3.    6a;  +  97/  +  12s!  =  5.  4.    lx  +  lly  +  Az  =  22. 

2x  +  3y  +  4.z  =  32.  2x  +  3y -{- 4:Z  =  9. 

3x-\-2y-z  =  8.  5x  +  2y-z  =  6. 

5.    lOx  +  5y —  15 z  =  5.         6.    2x  — 3y  —  4:Z  =  —  5. 
2x  +  y-3z  =  0.  x  +  6y  +  3z  =^10. 

3x-{-2  y  +  z  =  Q.  x  +  5y  +  3z  =  9. 


APPENDIX. 


415 


IX.  GRAPHS  OF  QUADRATIC  EQUATIONS  (p.  296). 

The  student  has  already  learned  in  Appendix  VIII  how 
graphically  to  represent  a  simple  equation.  Furthermore 
he  has  learned  that  to  every  point  on  the  graph  corresponds 
one  root  and  only  one  of  the  equation,  and  vice  versa,  a 
"one-to-one  correspondence"  between  points  and  roots. 

He  has  also  learned  that 


as,  in  general, 

two  straight  lines  in  a  plane 
have  one  common  point  and 
only  one, 

We  shall  now  consider  the  graphs 
of  equations  of  degrees  above  the 
first. 


so,  in  general, 

two    linear    equatio7is    have 

one   common  root  and  only 

one. 

Y 


Illustrative  problems.    1.  Required 
the  graph  of  the  equation 
a;2  +  7/2  ^  10. 


'•'  y  =  ±  vlO  —  x2,   .-.  by  giving  x  va- 
rious values  (noticing  that  x^  ^  10  for  real  values  of  y)  we  have  corres- 
ponding values  of  y  as  follows : 

x  =  ±  VIo,  ±  3,  ±  VS,  ±  Vt,  ±  VO,  ±  V5,       ±2,  ±  V2,  ±1,  0. 

y=  0,  ±  1,  ±  V2,  ±  Vs,      ±2,  ±  V5,  ±  Vo,  ±  Vs,  ±.3,  ±  VlO. 

Taking  the  approximate  square  roots,  and  laying  off  the  abscissas 
and  ordinates  as  indicated,  and  then  connecting  the  successive  points, 
the  graph  is  the  circumference  of  a  circle. 

So,  in  general,  the  graph  of  every  equation  of  the  forno  x^  +  y^  =  k^ 
is  the  circumference  of  a  circle. 

2.  Required  the  graph  of  the  equation  2  a;^  -f  3  y"^  =  10. 

■••  y  =  ±  iV6(5  —  x^),  .-.  by  giving  x  various  values  (noticing  that 
X-  ^  5  for  real  values  of  y)  we  have  corresponding  values  of  y  as  fol- 


lows: 


±V5,  ±2,       ±V3,    ±V2,  ±1,  ±0. 

0,  ±iV6,   ±fV3,   ±V2,  ±fV6,   iiVyo. 


416 


ELEMENTS   OF  ALGEBRA. 


Taking  the  approximate  square  roots,  and 
laying  off  the  abscissas  and  ordinates  as  indi- 
cated, and  then  connecting  the  successive  points, 
the  graph  is  a  curve  known  as  an  ellipse. 

So,  in  general,  the  graph  of  every  equation  of 
the  form  ax^  +  by2  =  c,  where  a,  b,  c  are  posi- 
tive, is  an  ellipse. 


3.  Required  the  graph  cf  the  equation  2x^  —  Sy^  =  10. 

'.'  y  =  ±-k "v^C (ic2  —  5),  .-.  by  giving  x  various  values  (noticing  that 
x2  <^  5  for  real  values  of  y)  we  have  corresponding  values  of  y  as  fol- 
lows: 

X  =  ±  V5,     ±  Ve,     ±  Vt,  ±  Vs, 

±iV6,    ±fV3,    ±V2, 


y 


0, 


±3, 
Y 


±  Vio, 


Taking  the  approximate 
square  roots,  and  laying  off 
the  abscissas  and  ordinates  as 
indicated,  and  then  connect- 
ing the  successive  points  the  X 
graph  is  the  curve  known  as 
the  hyperbola. 

So,  in  general,  the  graph 
of  every  equation  of  the  form 
ax2  —  by2  =  c,  where  a,  b,  c,  are  positive,  is  an  hyperbola. 

4.  Required  the  graph  of  the  equation  t/'^  =  Sx. 

'■'  y  —  ±i  "^2  X,  .-.  by  giving  x  various  values  (noticing  tl^at  x  <^  0 
for  real  values  of  y)  we  have  corresponding  values  for  y  as  follows  : 
x  =  0,  1,       2,  3,  4,  5,  6,  •••. 

y  =  0,   ±iV2,   ±1,   ±iV6,   ±V2,   iiVio,   ±V3,  .... 

Taking  the  approximate  square  roots,  and 
laying  off  the  abscissas  and  ordinates  as  indi- 
cated, and  connecting  the  successive  points,  the 
graph  is  a  curve  known  as  the  parabola. 

So,  in  general,  the  graph  of  every  equation 
of  the  form  y2  =  ax  is  a  parabola. 

The  ellipse,  hyperbola,  and  parabola  are  curves  formed  by  cutting 
into  a  right  circular  cone,  and  hence  are  called  conic  sections. 


APPENDIX.  417 

5.  Required  the  graph  of  the  equation 
cc^  +  3  ^2  -  a;  -  3  =  ?/. 

Giving  X  various  values,   we  have  corres- 
ponding values  of  y  as  follows : 

x  = 4,_3,_2,-l,      0,1,2,3,      4,--. 

y= 15,     0,     3,     0, -3,0,15,48, 105,.  ■.. 

The  curve  is  seen  to  be  one  which  can  be 
cut  by  a  straight  line  in  three  points,  and  this  is  the  general  charac- 
teristic of  graphs  of  cubic  equations. 

EXERCISES.    CLXXXVII. 

1.  Required  the  graph  of  the  equation  x'^-\-y^  —  '26.  In 
how  many  points  at  the  most  could  a  straight  line  cut  this 
curve  ? 

2.  Similarly  for  y^  =  18  aj. 

3.  Similarly  for  2x'^  +  5i/=^  10. 

4.  Similarly  for  2x''-^i/  =  10. 

5.  In  how  many  points,  at  the  most,  can  a  straight  line 
cut  the  graph  of  a  quadratic  equation,  judging  by  the  results 
of  exs.  1-4  ? 

6.  Required  the  graphs  of  the  equations 

x^  +  y'  =  13, 

drawn  with  respect  to  the  same  axes.  What  are  the  abcis- 
sas  and  ordinates  of  the  points  of  intersection  of  tl^e  two 
graphs  ?  How  do  these  compare  with  the  common  roots 
of  the  two  equations  ? 

7.  Similarly  for  the  equations 


418 


ELEMENTS   OF  ALGEBRA. 


8.  Similarly  for  the  equations 

f- 2x^  =  80. 

9.  Required  the  graph  of  the  equation  y  =  x^  —  9x, 
In  how  many  points  does  this  curve  cut  the  X  axis  ? 

10.    Required  the  graph  of  the  equation 
1/  =  x^  —  5  x^  -{-  9  x^  —  5  X. 
In  how  many  points  could  a  straight  line  cut  this  curve  ? 

Graphic  representation  of  the  roots.     In  the  equation 
y  =  x^  —  X  —  2 
we  have  the  following  corresponding  values  : 


3, 

-2, 

-1, 

0, 

1,   2,   3,     4,   . 

0, 

4, 

0, 

-2, 

-2,   0,    4,    10,    . 

The  graph  is  shown  in  the  annexed  figure." 
When   y  =  0,   x  =  —  1,  or  2.     That  is,   the 


values  of  x,  in  the  equation  x^ 


2  =  0, 


"^■^""X  are  the  abscissas  of  the  intersection  of  the^ 
graph  with  the  X  axis. 

Similarly,  any  equation  f(x)  =  0  can  be 
solved  by  writing  it  f(x)  =  y  and  plotting  it. 
The  abscissas  of  the  intersections  of  the  graph  with  the  X, 
axis  will  then  be  the  roots  of  the  equation. 

Imaginary  roots  show  themselves  by  a  curve  which  does 
not  reach  the  X  axis. 

JE.g.,  in  studying  the  equation  x^  —  5  x^  -^  8  x  —  &  =  0,  let 

y  =f(x).    Then  we  have  the  following  corresponding  values : 

x  =  ---    -1,        0,        1,  1.5,        2,    3,     4. 

y  = 20,    -6,    -2,    -1.875,    -2,   0,   10. 


APPENDIX. 


419 


The  curve  does  not  reach  the  X  axis  between  1  and  2. 
In  solvmg  the  equation  the  roots  are  found  to  be  1  +  i, 
1  —  i,  and  3. 

The  fact  that  complex  roots  enter  in  pairs 
is  readily  understood  by  a  study  of  the  graph. 

E.g.,  consider  the  equation  ic^  —  4  =  0. 

Let  f{x)  =  y.     We  then  have  the  following 
corresponding  values  : 

x=      0,    ±1,    ±2,    ±3,    .... 
2/--4,    -3,        0,        5,    .... 

If,  now,  we  make  each  y  3  units  greater,  i.e.,  if  we  lift 
the  curve  3  units  (or,  what  is  the  same  thing,  lower  the  X 
axis  3  units)  x  will  equal  ±  1  when  y  =  0. 
I.e.,  the  roots  will  approach  each  other. 
This  can  be  done  by  making  the  equation 
x''-l=y. 

If  we  lift  the  curve  another  unit  (or  lower 
the  X  axis  4  units),  making  the  equation 
x^  =  y,  X  will  have  only  the  double  root  0 
when  y  =  0;  i.e.,  the  two  roots  are  now  equal. 

If  we  lift  the  curve  another  unit  (or  lower  the  X  axis  5 
units),  making  the  equation  x^-\-.l  =  y,  x  will  have  the 
imaginary  value  ±  i  when  y  =  0,  the  two  imaginaries  enter- 
ing together.  In  other  words,  complex  roots  (of  which  pure 
imaginaries  form  a  special  class)  enter  in  pairs. 

Roots  of  simultaneous  equations.  We  have  seen  that  two 
linear  equations,  involving  but  two  unknown  quantities,  can 
be  solved  by  finding  the  point  of  intersection  of  their  graphs. 

Similarly,  if  we  have  two  equations  like 


I. 
11. 


x'  +  3i/  =  2S, 
2x^-y^  =  -7, 


420 


ELEMENTS   OF   ALGEBRA. 


the  coordinates  of  the  common  points  of  their  graphs  repre- 
sent the  common  roots  of  the  equations. 

From  I,  we  have  y  =  ±^  V3(28  —  x^). 

Hence,  if 

x=     0,  ±  1,  ±  2,         ±3,  ±  4,  ±  5,  ±  V28, 

then     2/  =  ±fV2l,  ±3,  ±2V2,  ±iV57,  ±2,  ±1,       0. 


The  graph  is  marked  I  in  the  annexed  figure.  It  is  an 
ellipse. 

From  II,  we  have  3/  =  ±  V2  x^  +  7. 

Hereifcc=      0,       ±  1,  ±  2,         ±  3,  ±  4,         ••., 

then  7/  =  ±  V7,  ±3,  ±  Vl5,    ±  5,  ±  V39,   ■  •  •. 

The  graph  is  marked  II  in  the  annexed  figure.     It  is  an 
hyperbola,  a  two-branched  figure. 
The  common  roots  are 

x  =  l,        1,-1,-1. 

y  =  3,    -3,        3,    -3. 

From  the  preceding  figure  we  see  confirmed  the  fact 
already  mentioned,  that  two  simultaneous  quadratic  equa- 
tions involving  two  unknown  quantities  have,  in  general, 
four  roots,  the  two  curves  intersecting  in  four  points. 

Two  of  the  points  may  coincide,  as  in  Fig.  1,  giving  a 
double  root,  or  there  may  be  two  double  roots,  as  in  Fig.  2, 


APPENDIX. 


421 


or  two  of  the  roots  may  be  imaginary,  as  in  Fig.  3,  or  both 
pairs  of  roots  may  be  imaginary. 


Fio.  1. 


Fig.  2. 


Fig.  3. 


From  similar  considerations  it  may  be  inferred  that  there 
are  6  roots  common  to  two  simultaneous  equations  of  which 
one  is  a  quadratic  and  the  other  a  cubic.  In  general,  if  one 
equation  is  of  the  Mth  degree  and  the  other  of  the  fith, 
there  are  m?i  roots. 


EXERCISES.    CLXXXVIII. 

Represent  graphically  the  following  sets  of  simultaneous 
equations,  and  find  at  least  one  value  of  x  and  one  of  i/ 
common  to  the  two. 


1.    x^  +  y^  =  S. 

2.    x'^y  =  7. 

y^  =  2x. 

x  +  y'  =  ll. 

3.    a; +  37/ =  10. 

4.    2.^2^7/2  =  19. 

x'-h5  =  2y. 

x^-y^  =  8. 

5.    2x-'  +  y^  =  3. 
ic2  _  3  7/2  =  _  2. 


6.    x^  +  x^  —  x  —  3  =  y. 
x  +  y  =  5. 


7.    a;2  +  7/2  =  5. 


y  =  x^  +  3x^-4:X-\-2. 


TABLE    OF    BIOGRAPHIES. 


The  simple  equation  was  known  to  the  Egyptians,  its  solution 
appearing  in  the  oldest  deciphered  mathematical  work  extant,  the 
papyrus  of  Ahnies.  The  quadratic  equation  was  solved  by  the 
Greeks,  and  indeterminate  equations  formed  a  considerable  portion 
of  the  works  of  Diophantus.  The  Hindus,  Persians,  and  Arabs  next 
took  up  the  science  and  made  considerable  progress  in  the  study  of 
equations  and  series.     The  Arabs  gave  to  algebra  its  name. 

The  sixteenth  century  saw  a  great  revival  of  learning  in  general 
and  of  algebra  in  particular.  The  cubic  and  quartic  equations  were 
now  solved. 

The  seventeenth  century  saw  modern  symbolism  established,  thus 
forming  elementary  algebra  as  it  is  known  to-day. 

The  following  table  contains  the  names  mentioned  in  this  work, 
together  with  a  few  others  prominent  in  the  history  of  algebra.  The 
notes  are  chiefly  from  those  prepared  by  the  authors  for  their  transla- 
tion of  Fink's  "  History  of  Mathematics"  (Chicago,  The  Open  Court 
Publishing  Co.,  1900),  to  which  reference  is  made  for  a  more  complete 
account  of  the  development  of  the  science. 

Abel,  Niels  Henrik.  Born  at  Pindoe,  Norway,  August  6,  1802  ;  died 
April  6,  1829.  Proved  the  impossibility  of  the  algebraic  solution 
of  the  quintic  equation. 

Ahmes.  An  Egyptian  scribe.  Lived  about —1700.  Wrote  the  earliest 
deciphered  mathematical  manuscript  extant,  on  arithmetic,  alge- 
bra, and  mensuration. 

Al  Khowarazmi,  Abu  Jafar  Mohammed  ibn  Musa.  First  part  of  ninth 
century.  Native  of  Khwarazm  (Khiva).  Arab  mathematician 
and  astronomer.     The  title  of  his  work  gave  the  name  to  algebra. 

Ampere,  Andr^-Marie.      Born  at  Lyons,  France,  in  1775 ;   died  at 
Marseilles  in  1836.     Founder  of  the  science  of  electro-dynamics. 
423 


424  TABLE   OF  BIOGRAPHIES. 

Apollonius  of  Perga,  in  Pamphylia.  Taught  at  Alexandria  between 
—  250  and  —  200,  in  the  reign  of  Ptolemy  Philopator.  Solved 
.the  general  quadratic  with  the  help  of  conies. 

Aigand,  Jean  Robert.  Born  at  Geneva,  1768 ;  died  c.  1825.  Private 
life  unknown.  One  of  the  inventors  of  the  present  method  of 
geometrically  representing  complex  numbers  (1806). 

Aristotle.  Born  at  Stageira,  Macedonia,  —  384 ;  died  at  Chalcis, 
Eubcea,  —  322.  Great  philosopher.  Represented  unknown  quan- 
tities by  letters ;  suggested  the  theory  of  combinations. 

Aryabhatta.  Born  at  Pataliputra  on  the  Upper  Ganges,  476.  Hindu 
mathematician.  Wrote  on  algebra,  including  quadratic  equations, 
permutations,  indeterminate  equations,  and  magic  squares. 

B^zout,  Etienne.  Born  at  Nemours  in  1730 ;  died  at  Paris  in  1783. 
Prominent  in  the  study  of  symmetric  functions  and  determinants. 

Bhaskara  Acharya.  Born  in  1114.  Hindu  mathematician  and  astrono- 
mer. Author  of  the  "  Lilavati "  and  the  "  Vijaganita,"  contain- 
ing the  elements  of  arithmetic  and  algebra.  One  of  the  most 
prominent  mathematicians  of  his  time. 

Bombelli,  Rafaele.  Italian.  Born  c.  1530.  His  algebra  (1572)  sum- 
marized all  then  known  on  the  subject,  especially  the  cubic. 

Boyle,  Robert.  Born  in  Ireland,  Jan.  25,  1627 ;  died  Dec.  30,  1691. 
Celebrated  physicist. 

Brahmagupta.  Born  in  598.  Hindu  mathematician.  Contributed  to 
geometry,  trigonometry,  and  algebra. 

Briggs,  Henry.  Born  at  Warley  Wood,  near  Halifax,  Yorkshire, 
February,  1560-1 ;  died  at  Oxford  Jan.  26,  1630-1.  Savilian  pro- 
fessor of  geometry  at  Oxford.  Among  the  first  to  recognize  the 
value  of  logarithms ;   those  with  decimal  base  bear  his  name. 

Burgi,  Joost  (Jobst).  Born  at  Lichtensteig,  St.  Gall,  Switzerland, 
1552  ;  died  at  Cassel  in  1632.  One  of  the  first  to  suggest  a  system 
of  logarithms.  The  first  to  recognize  the  value  of  making  the 
second  member  of  an  equation  zero. 

Cardan,  Jerome  (Hieronymus,  Girolamo).  Born  at  Pa  via,  1501 ;  died 
at  Rome,  1576.  Professor  of  mathematics  at  Bologna  and  Padua. 
Mathematician,  physician,  astrologer.  First  to  publish  (1545)  the 
solution  of  the  cubic  equation. 

Cataldi,  Pietro  Antonio.  Italian  mathematician,  born  1548  ;  died  at 
Bologna,  1626.  Professor  of  mathematics  at  Florence,  Perugia, 
and  Bologna.     Pioneer  in  the  use  of  continued  fractions. 


I 


TABLE  OF  biographip:s.  425 

Cauchy,  Augustin  Louis.  Born  at  Paris,  1789  ;  died  at  Sceaux,  1857. 
Professor  of  niatliematics  at  Paris.  One  of  tlie  most  prominent 
mathematicians  of  his  time.  Contributed  to  the  theory  of  deter- 
minants, series,  and  algebra  in  general. 

Cramer,  Gabriel.  Born  at  Geneva,  1704 ;  died  at  Bagnols,  1752. 
Added  to  the  theory  of  equations  and  revived  the  study  of  deter- 
minants (begun  by  Leibnitz). 

D'Alembert,  Jean  le  Rond.  Born  at  Paris,  1717;  died  there,  1783. 
Physicist,  mathematician,  astronomer.  Contributed  to  the  theory 
of  equations. 

De  Moivre,  Abraham.  Born  at  Vitry,  Champagne,  1667  ;  died  at 
London,  1754.  Contributed  to  the  theory  of  complex  numbers 
and  of  probabilities. 

Descartes,  Ren^,  du  Perron.  Born  at  La  Haye,  Touraine,  1596  ;  died 
at  Stockholm,  1650.  Discoverer  of  analytic  geometry.  Contrib- 
uted extensively  to  algebra. 

Diophantus  of  Alexandria.  Lived  about  275.  Most  prominent  of 
Greek  algebraists,  contributing  to  indeterminate  equations. 

Euclid.  Lived  about  —  300.  Taught  at  Alexandria  in  the  reign  of 
Ptolemy  Soter.  The  author  or  compiler  of  the  most  famous  text- 
book of  geometry  ever  vv^ritten,  the  "Elements,"  in  thirteen  books. 

Euler,  Leonhard.  Born  at  Basel,  1707  ;  died  at  St.  Petersburg,  1783. 
One  of  the  greatest  physicists,  astronomers,  and  mathematicians  of 
the  eighteenth  century. 

Ferrari,  Ludovico.  Born  at  Bologna,  1522  ;  died  in  1562.  Solved  the 
biquadratic. 

Ferro,  Scipione  del.  Born  at  Bologna,  c.  1465  ;  died  between  Oct.  29 
and  Nov.  16,  1526.  Professor  of  mathematics  at  Bologna.  Inves- 
tigated the  geometry  based  on  a  single  setting  of  the  compasses, 
and  was  the  first  to  solve  the  special  cubic  x^  +  px  =  q. 

Gauss,  Karl  Friedrich.  Born  at  Brunswick,  1777  ;  died  at  Gottingen, 
1855.  The  greatest  mathematician  of  modern  times.  Prominent 
as  a  physicist  and  astronomer.  The  theories  of  numbers,  of  func- 
tions, of  equations,  of  determinants,  of  complex  numbers,  of 
hyperbolic  geometry,  are  all  largely  indebted  to  his  great  genius. 

Harriot,  Thomas.  Born  at  Oxford,  1560 ;  died  at  Siou  House,  near 
Isleworth,  July  2,  1621.  The  most  celebrated  English  algebraist 
of  his  time. 


426  TABLE   OF  BIOGRAPHIES. 

Horner,  William  George.  Born  in  1786 ;  died  at  Bath,  Sept.  22,  1837. 
Chiefly  known  for  his  method  of  approximating  the  real  roots 
of  a  numerical  equation  (1819). 
Lagrange,  Joseph  Louis,  Comte.  Born  at  Turin,  Jan.  25,  1736;  died 
at  Paris,  April  10,  1813.  One  of  the  foremost  mathematicians  of 
his  time.  Contributed  extensively  to  the  calculus  of  variations, 
theory  of  numbers,  determinants,  and  theory  of  equations. 

Maclaurin,  Colin.  Born  at  Kilmodan,  Argyllshire,  1698  ;  died  at  York, 
June  14,  1746.  Professor  of  mathematics  at  Edinburgh.  Contrib- 
uted to  the  study  of  conies  and  series. 

Metrodorus.     Lived  about  325,     Author  of  many  algebraic  problems. 

Napier,  John.  Born  at  Merchiston,  then  a  suburb  of  Edinburgh,  1550 ; 
died  there  in  1617.     Inventor  of  logarithms. 

Newton,  Sir  Isaac.  Born  at  Woolsthorpe,  Lincolnshire,  Dec.  25,  1642, 
(0.  S.);  died  at  Kensington,  March  20,  1727.  Lucasian  professor 
of  mathematics  at  Cambridge  (1669).  The  world's  greatest  mathe- 
matical physicist.  Invented  fluxional  calculus  (c.  1666).  Contrib- 
uted extensively  to  the  theories  of  series,  equations,  curves,  etc. 

Ohm,  Georg  Simon.  Born  at  Erlangen,  Germany,  in  1781 ;  died  July 
6;  1854.     Celebrated  physicist. 

Pascal,  Blaise.  Born  at  Clermont,  1623  ;  died  at  Paris,  1662.  Physi- 
cist, philosopher,  mathematician.  Contributed  to  the  theory  6t 
numbers,  probabilities,  and  geometry. 

Recorde,  Robert.  Born  at  Tenby,  Wales,  1510 ;  died  in  prison,  at 
London,  1558.  Professor  of  mathematics  and  rhetoric  at  Oxford. 
Introduced  the  sign  =  for  equality. 

Tartaglia,  Nicolo.  (Nicholas  the  stammerer.  Real  name,  Nicolo  Fon- 
tana.)  Born  at  Brescia,  c.  1500;  died  at  Venice,  c.  1557.  Physi- 
cist and  arithmetician ;  known  for  his  work  on  cubic  equations. 

Taylor,  Brook.  Born  at  Edmonton,  1685 ;  died  at  London,  1731. 
Physicist  and  mathematician.      Known  for  his  work  in  series. 

Vifete  (Vieta),  Frangois.  Born  at  Fontenay-le-Comte,  1540 ;  died  at 
Paris,  1603.     The  foremost  algebraist  of  his  time. 

Volta,  Alessandro.  Born  at  Como,  Italy,  Feb.  18,  1745  ;  died  March 
5,  1827.     Celebrated  physicist. 

Wallis,  John.  Born  at  Ashford,  1616  ;  died  at  Oxford,  1703.  Savilian 
professor  of  geometry  at  Oxford.  Suggested  (1685)  the  modem 
graphic  interpretation  of  the  imaginary. 


TABLE    OF   ETYMOLOGIES. 


The  following  table  will  serve  to  make  more  clear  to  students  the 
meaning  of  many  words  used  and  defined  in  elementary  algebra, 
i.,  Latin.  G.,  Greek.  dim.,  diminutive. 


Abscissa.     L.  cut  off. 

Absolute.  L.  absolutus,  ab,  from, 
+  solvere^  loosen.  That  is,  com- 
plete, unrestricted. 

Abstract.  L.  abs,  away,  +  tra- 
here,  draw. 

Add.  L.  ad,  to,  +  -dere,  for  dare, 
put,  place. 

Affected.  L.  ad,  to,  +facere,  do, 
make;  i.e.,  to  act  upon,  influ- 
ence. Hence  compounded ;  an 
equation  of  several  degrees. 

Aggregation.  L,  ad,  to,  -f  gre- 
gare,  collect  into  a  flock,  ^from 
grex,  flock. 

Algebra.  Arabic,  al,  the,  -\-jabr, 
redintegration,  consolidation. 
The  title  of  Al  Khowarazmi's 
work  (see  Table  of  Biographies) 
was  Him  al-jabr  wa''  I  muqa- 
balah,  the  science  of  redintegra- 
tion and  equation ;  of  this  long 
title  only  al-jabr  survives. 

Alternation.     L.  alter,  other. 

Antecedent.  L.  ante,  before,  -|- 
cedere,  go. 

Antilogarithm.  L.  and  G.  anti-, 
against,    opposite  to,    +  loga- 


rithm. See  Logarithm.  The 
number  standing  opposite  to  the 
logarithm. 

Arithmetic.  G.  arithmos,  num- 
ber. 

Ascend.  L.  ad-,  to,  +  scandere, 
climb. 

Associative.  L.  ad-,  to, -f-sociare, 
join. 

Axiom.  G.  axioma,  that  which 
is  thought  fit,  a  requisite. 

Binomial.  L.  bi-,  two-,  +  nomen, 
name. 

Characteristic.  G.  charakterizein, 
designate;  from  G.  character, 
an  instrument  for  graving,  from 
charassein,  to  scratch. 

Circulate.  From  circle.  L.  dim. 
of  circus,  a  ring,  G.  kirkos  or 
krikos,  a  circle,  ring. 

Commutative.  L.  com-,  inten- 
sive, -I-  mutare,  change. 

Comparison.  L.  com-,  together 
with,  +  par,  equal. 

Complement.  L.  complementum, 
that  which  fills ;  from  com-, 
intensive,  +  plere,  fill. 


427 


428 


TABLE   OF   ETYMOLOGIES. 


Complete.     See  Complement. 
Complex.     L.  com.,  together,  + 

plectere,  weave. 
Composition.     L.  com-,  together, 

4-  ponere,  place. 
Cjompound.     Same  etymology  as 

Composition. 
Consequent.      L.  con-,  together, 

+  sequi,  follow. 
Constant.     L.  con-,  together,  -f 

stare,  stand. 
Continued.     L.  con-,  together,  + 

tenere,  hold. 
Corollary.     L.  corollarium,  a  gift, 

money  paid  for   a  garland  of 

flowers,  from  corolla,  dim.  of 

corona,  a  crown. 
Cube.     G.  cuhos,  a  die,  a  cube. 


Decimal.     L.  decem,  ten. 
Deduce.     L.  de,  down,  away,  + 

ducere,  lead. 
Define.      L.  de-,    +finire,  limit, 

settle,  define. 
Degree.     L.  de,  down,  +  gradus, 

step. 
Denominator.      L.   namer,   from 

de,    +  nominare,   name,  from 

nomen,  name. 
Descend.     L.  de,  down,   +  scan- 

dere,  climb. 
Detach.     Ital.  des-,  privative,  + 

-tacher,  fasten. 
Determinant.      L.  de-,   +  termi- 

nare,  bound,  limit. 
Determine.     See  Determinant. 
Discriminant.     L.  dis-,  apart,  + 

cernere  =  G.  krinein,  separate. 
Distribute.     L.  dis-,  apart,  +  tri- 

buere,  give. 


Divide.     L.  di-,  for  dis-,  apart,  -|- 

videre,  see. 
Domain,  L.dommmm,  dominion, 

from  dominus,  lord. 

Eliminate.  L.  e,  out,  +  limen,  a 
threshold.  To  turn  out  of  doors. 

Equal,  L.  aequalis,  equal,  from 
aequus,  plain. 

Equation.     See  Equal. 

Evolution.  L.  e,  out,  -|-  volvere, 
roll.     To  unfold  the  root. 

Exponent.  L.  ex,  out,  +  ponere, 
put ;  i.e.,  to  set  forth,  indicate. 

Extraneous.     L.  extra,  outside. 

Extreme.  L.  extremus,  superla- 
tive of  exter,  outer. 

Factor.     L.  a  doer,  from  facere, 

do. 
Fraction,      L.   fractus,    broken, 

from,  frangere,  break. 
Function.    L,  functus,  performed, 

from,  fungi,  perform. 

Graph.     G.  graphein,  write. 

Homogeneous.  G.  homos,  the 
same,  +  genos,  race. 

Identical.     L,  idem,  the  same. 

Imaginary.     L.  imago,  an  image. 

Indeterminate,  L,  in-,  privative, 
4-  determinate.  See  Determi- 
nants. 

Index,  L,  indicare,  point  out, 
show. 

Infinite,  L,  in-,  not,  +Jinitus, 
bounded. 

Inspection.  L.  in,  on,  in,  at,  + 
specere,  look. 


TABLE   OF   ETYMOLOGIES. 


429 


Integer.     L.  in,  privative,  +  tan- 

gere,   toucli;    i.e.,   untouched, 

whole,  sound. 
Inverse.      L.   in,  on,  toward,  + 

vertere,  turn. 
Involution.     L.  in,  in,  +  volvere, 

roll.     To  roll  the  root  into  a 

power. 

Limit.  L.  limes  (limit-),  a  cross- 
path,  boundary. 

Linear.     L.  linea,  line. 

Literal.     L.  littera,  litera,  a  letter. 

Logarithm.  G.  logos,  proportion, 
ratio,  +  arithmos,  number. 

Mantissa.     L.  an  addition. 

Maximum.  L.  greatest,  superla- 
tive of  magnus,  great. 

Mean.     L.  medius,  middle. 

Minimum.     L.  least. 

Minuend.     L.  minuere,  lessen. 

Monomial.  G.  monos,  single, 
+  L.  nomen,  name. 

Multiple.  L.  multus,  many,  + 
-plus,  like  English  -fold,  from 
plicare,  fold. 

Negative.     L.  ne,  not,  +  que,  a 

generalizing  suffix. 
Notation.     L.  notatio,  a  marking, 

from  nota,  a  mark,  a  sign. 
Number.     L.  numerus,  number. 
Numerator.     L.  numberer. 

Operation.     L.  opus,  work. 
Ordinate.    L.  ordo  (ordin-),  a  row. 

7t.  Initial  of  G.  periphereia,  pe- 
riphery, circumference. 


Polynomial.     G.  polus,  many,  + 

L.  nomen,  name. 
Positive.      L.  positivus,   settled, 

from  ponere,  put. 
Power.     L.  posse,  to  be  able. 
Problem.    G.  problema,  a  question 

proposed    for    solution ;    from 

pro,  before,   -|-  ballein,  throw. 
Product.     L.  pro-,  forth,   +  du- 

cere,  lead. 
Proportion.     L.  pro,  for,  before, 

+  portio,  a  share. 
Proposition.      L.  pro,  before,  + 

ponere,  place. 
Pure.     L.  purus,  clean. 

Quadratic.  L.  quadratus,  a 
square,  from  quattuor,  four. 

Quantity.  L.  quantus,  how  much, 
from  quam,  how. 

Quartic.     L.  quattuor,  four. 

Quotient.     L.  quot,  how  many. 

Radical.     L.  radix,  root. 

Ratio.  L.  a  reckoning,  calcula- 
tion, from  reri,  think,  estimate. 

Rational.     L.  ratio.     See  Ratio. 

Real.  L.  realis,  belonging  to  the 
thing  itself,  from  res,  thing. 

Reciprocal.  L.  re-,  back,  +  ad- 
jective formative  -cus. 

Reduce.  L.  re-,  back,  +  ducere, 
lead. 

Remainder.  L.  re-,  behind,  back, 
-I-  manere,  remain. 

Root.     L.  and  G.  radix,  a  root. 

Series.     L.  a  row. 
Similar.     L.  similis,  like. 
Simplify.     L.  simplex,  simple. 


430 


TABLE   OF  ETYMOLOGIES. 


Simultaneous,  L.  simultim,  at 
the  same  time,  from  simul, 
together. 

Solution.     L.  solvere,  loose. 

Square.  L.  quadra,  a  square, 
from  quattuor,  four. 

Substitute.  L.  sub,  under,  + 
statuere,  set  up. 

Subtract.  L.  sub,  +trahere,  draw. 

Subtrahend.     See  Subtract. 

Sum.     L.  sumnia,  highest  part. 

Surd.  L.  surdus,  deaf.  A  mis- 
translation of  the  G.  alogos, 
which  does  not  mean  stupid 
(hence  deaf),  but  inexpressible. 


Symbol.  G.  symbolos,  a  mark, 
from  syn,  together,  +  ballein, 
put. 

Symmetry.  G.  syn,  together,  + 
metron,  measure. 

Theorem.  G.  theorema,  a  sight, 
a  principle  contemplated. 

Transpose.  L.  trans,  over,  + 
ponere,  place. 

Trinomial.  L.  tres  (tri-),  three, 
+  nomen,  name. 

Vary,  variation.  L.  varius,  dif- 
ferent. 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

LOAN  DEPT. 

This  book  is  due  on  the  last  date  stamped  below,  or 

on  the  date  to  which  renewed. 

Renewed  books  are  subjea  to  immediate  recall. 


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